Practice · Let the sign decide the exponent's parity · Sensim Math

Variant 1 easy

In the equation $\left(-\frac{x^3}{y}\right)^a\times\left(\frac{y^2}{x^b}\right)^3\div\left(-\frac{x^2}{2y}\right)^2=\frac{4y^c}{x}$ , the numbers $a$ , $b$ , and $c$ are natural numbers and $1\le a\le3$ . Find $a+b+c$ .

Your answer

Show solution

1 · Understandwhat's really being asked

An equation between monomial expressions contains unknown natural-number exponents $a$ , $b$ , $c$ with $1\le a\le3$ . Find $a+b+c$ .

Givens

$\left(-\frac{x^3}{y}\right)^a\times\left(\frac{y^2}{x^b}\right)^3\div\left(-\frac{x^2}{2y}\right)^2=\frac{4y^c}{x}$
$a$ , $b$ , $c$ are natural numbers.
$1\le a\le3$

Unknowns

$a+b+c$

Constraints

Both sides must agree in sign of the coefficient and in the exponent of each letter.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#7 Identify Subproblems

The sign of the left side depends only on whether $a$ is even or odd, so settle $a$ first by matching signs. With $a$ known, simplify the left side and match the exponents of $x$ and $y$ .

3 · Execute6 carry out the plan

1Look at the sign first

#3 Eliminate Possibilities 7.NS.A.2

\left(\frac{y^2}{x^b}\right)^3

has no minus sign and

\left(-\frac{x^2}{2y}\right)^2

is an even power, so only

\left(-\frac{x^3}{y}\right)^a

can make the coefficient negative. The coefficient on the left is

(-1)^a\div\frac{1}{4}

.

(-1)^a\times4=4 \Rightarrow (-1)^a=1

A negative number to an odd power stays negative; to an even power it turns positive.

2Pin down $a$

#3 Eliminate Possibilities 7.NS.A.2

a

must be even and

1\le a\le3

, which rules out

1

and

3

.

a=2

Only one even number fits in the range.

3Expand each power

#7 Identify Subproblems 8.EE.A.1

Raise every factor inside each parenthesis to its power.

\left(-\frac{x^3}{y}\right)^2=\frac{x^6}{y^2},\ \left(\frac{y^2}{x^b}\right)^3=\frac{y^6}{x^{3b}},\ \left(-\frac{x^2}{2y}\right)^2=\frac{x^4}{4y^2}

(x^3)^2=x^6

,

(y^2)^3=y^6

,

(2y)^2=4y^2

.

4Turn the division into multiplication

#13 Convert to Algebra 8.EE.A.1

Dividing by

\frac{x^4}{4y^2}

means multiplying by

\frac{4y^2}{x^4}

. Then collect powers of

x

and

y

.

\frac{x^6}{y^2}\times\frac{y^6}{x^{3b}}\times\frac{4y^2}{x^4}=\frac{4y^6}{x^{3b-2}}

For

x

:

6-3b-4=2-3b

. For

y

:

-2+6+2=6

.

5Match exponents

#13 Convert to Algebra 8.EE.A.1

Compare with

\frac{4y^c}{x}

: the powers of

y

and of

x

must agree.

c=6,\quad 3b-2=1 \Rightarrow b=1

Same letter, same exponent on both sides.

6Add

#13 Convert to Algebra 8.EE.A.1

Add the three values.

a+b+c=2+1+6=9

All three are natural numbers, as required.

Answer:

9

4 · Reviewdoes it hold up?

Put $a=2$ , $b=1$ back in: $\frac{x^6}{y^2}\times\frac{y^6}{x^3}\times\frac{4y^2}{x^4}=4x^{6-3-4}y^{-2+6+2}=\frac{4y^6}{x}$ , which is the right side with $c=6$ .

Another way: Match exponents first: for $x$ , $3a-3b-4=-1$ ; for $y$ , $-a+6+2=c$ . In the range this allows $(a,b)=(2,1)$ or $(a,b)=(3,2)$ . The sign then rules out $a=3$ (odd power, negative coefficient), leaving $a=2$ , $b=1$ , $c=6$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Expanding powers of quotients and collecting exponents of $x$ and $y$.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Deciding the sign of a power of a negative number.

💡Takeaway. Before you chase exponents, check the sign -- it can settle an unknown in one glance.

Variant 2 easy

In the equation $\left(-\frac{x^4}{y}\right)^a\times\left(\frac{y}{x^b}\right)^2\div\left(-\frac{x}{3y}\right)^2=\frac{9y^c}{x^2}$ , the numbers $a$ , $b$ , and $c$ are natural numbers and $1\le a\le3$ . Find $a+b+c$ .

Your answer

Show solution

1 · Understandwhat's really being asked

An equation between monomial expressions contains unknown natural-number exponents $a$ , $b$ , $c$ with $1\le a\le3$ . Find $a+b+c$ .

Givens

$\left(-\frac{x^4}{y}\right)^a\times\left(\frac{y}{x^b}\right)^2\div\left(-\frac{x}{3y}\right)^2=\frac{9y^c}{x^2}$
$a$ , $b$ , $c$ are natural numbers.
$1\le a\le3$

Unknowns

$a+b+c$

Constraints

Both sides must agree in sign of the coefficient and in the exponent of each letter.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#7 Identify Subproblems

The sign of the left side depends only on whether $a$ is even or odd, so settle $a$ first by matching signs. With $a$ known, simplify the left side and match the exponents of $x$ and $y$ .

3 · Execute6 carry out the plan

1Look at the sign first

#3 Eliminate Possibilities 7.NS.A.2

\left(\frac{y}{x^b}\right)^2

has no minus sign and

\left(-\frac{x}{3y}\right)^2

is an even power, so only

\left(-\frac{x^4}{y}\right)^a

can make the coefficient negative. The coefficient on the left is

(-1)^a\div\frac{1}{9}

.

(-1)^a\times9=9 \Rightarrow (-1)^a=1

A negative number to an odd power stays negative; to an even power it turns positive.

2Pin down $a$

#3 Eliminate Possibilities 7.NS.A.2

a

must be even and

1\le a\le3

, which rules out

1

and

3

.

a=2

Only one even number fits in the range.

3Expand each power

#7 Identify Subproblems 8.EE.A.1

Raise every factor inside each parenthesis to its power.

\left(-\frac{x^4}{y}\right)^2=\frac{x^8}{y^2},\ \left(\frac{y}{x^b}\right)^2=\frac{y^2}{x^{2b}},\ \left(-\frac{x}{3y}\right)^2=\frac{x^2}{9y^2}

(x^4)^2=x^8

,

(3y)^2=9y^2

.

4Turn the division into multiplication

#13 Convert to Algebra 8.EE.A.1

Dividing by

\frac{x^2}{9y^2}

means multiplying by

\frac{9y^2}{x^2}

. Then collect powers of

x

and

y

.

\frac{x^8}{y^2}\times\frac{y^2}{x^{2b}}\times\frac{9y^2}{x^2}=\frac{9y^2}{x^{2b-6}}

For

x

:

8-2b-2=6-2b

. For

y

:

-2+2+2=2

.

5Match exponents

#13 Convert to Algebra 8.EE.A.1

Compare with

\frac{9y^c}{x^2}

: the powers of

y

and of

x

must agree.

c=2,\quad 2b-6=2 \Rightarrow b=4

Same letter, same exponent on both sides.

6Add

#13 Convert to Algebra 8.EE.A.1

Add the three values.

a+b+c=2+4+2=8

All three are natural numbers, as required.

Answer:

8

4 · Reviewdoes it hold up?

Put $a=2$ , $b=4$ back in: $\frac{x^8}{y^2}\times\frac{y^2}{x^8}\times\frac{9y^2}{x^2}=9x^{8-8-2}y^{-2+2+2}=\frac{9y^2}{x^2}$ , which is the right side with $c=2$ .

Another way: Match exponents first: for $x$ , $4a-2b-2=-2$ ; for $y$ , $-a+2+2=c$ . In the range this allows $(a,b)=(1,2)$ or $(a,b)=(2,4)$ or $(a,b)=(3,6)$ . The sign then rules out $a=1$ and $a=3$ (odd power, negative coefficient), leaving $a=2$ , $b=4$ , $c=2$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Expanding powers of quotients and collecting exponents of $x$ and $y$.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Deciding the sign of a power of a negative number.

💡Takeaway. Before you chase exponents, check the sign -- it can settle an unknown in one glance.

Variant 3 easy

In the equation $\left(-\frac{x^2}{y}\right)^a\times\left(\frac{y^2}{x^b}\right)^2\div\left(-\frac{x}{2y^2}\right)^2=-\frac{4y^c}{x^2}$ , the numbers $a$ , $b$ , and $c$ are natural numbers and $2\le a\le4$ . Find $a+b+c$ .

Your answer

Show solution

1 · Understandwhat's really being asked

An equation between monomial expressions contains unknown natural-number exponents $a$ , $b$ , $c$ with $2\le a\le4$ . Find $a+b+c$ .

Givens

$\left(-\frac{x^2}{y}\right)^a\times\left(\frac{y^2}{x^b}\right)^2\div\left(-\frac{x}{2y^2}\right)^2=-\frac{4y^c}{x^2}$
$a$ , $b$ , $c$ are natural numbers.
$2\le a\le4$

Unknowns

$a+b+c$

Constraints

Both sides must agree in sign of the coefficient and in the exponent of each letter.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#7 Identify Subproblems

The sign of the left side depends only on whether $a$ is even or odd, so settle $a$ first by matching signs. With $a$ known, simplify the left side and match the exponents of $x$ and $y$ .

3 · Execute6 carry out the plan

1Look at the sign first

#3 Eliminate Possibilities 7.NS.A.2

\left(\frac{y^2}{x^b}\right)^2

has no minus sign and

\left(-\frac{x}{2y^2}\right)^2

is an even power, so only

\left(-\frac{x^2}{y}\right)^a

can make the coefficient negative. The coefficient on the left is

(-1)^a\div\frac{1}{4}

.

(-1)^a\times4=-4 \Rightarrow (-1)^a=-1

A negative number to an odd power stays negative; to an even power it turns positive.

2Pin down $a$

#3 Eliminate Possibilities 7.NS.A.2

a

must be odd and

2\le a\le4

, which rules out

2

and

4

.

a=3

Only one odd number fits in the range.

3Expand each power

#7 Identify Subproblems 8.EE.A.1

Raise every factor inside each parenthesis to its power.

\left(-\frac{x^2}{y}\right)^3=-\frac{x^6}{y^3},\ \left(\frac{y^2}{x^b}\right)^2=\frac{y^4}{x^{2b}},\ \left(-\frac{x}{2y^2}\right)^2=\frac{x^2}{4y^4}

(x^2)^3=x^6

,

(y^2)^2=y^4

,

(2y^2)^2=4y^4

.

4Turn the division into multiplication

#13 Convert to Algebra 8.EE.A.1

Dividing by

\frac{x^2}{4y^4}

means multiplying by

\frac{4y^4}{x^2}

. Then collect powers of

x

and

y

.

-\frac{x^6}{y^3}\times\frac{y^4}{x^{2b}}\times\frac{4y^4}{x^2}=-\frac{4y^5}{x^{2b-4}}

For

x

:

6-2b-2=4-2b

. For

y

:

-3+4+4=5

.

5Match exponents

#13 Convert to Algebra 8.EE.A.1

Compare with

-\frac{4y^c}{x^2}

: the powers of

y

and of

x

must agree.

c=5,\quad 2b-4=2 \Rightarrow b=3

Same letter, same exponent on both sides.

6Add

#13 Convert to Algebra 8.EE.A.1

Add the three values.

a+b+c=3+3+5=11

All three are natural numbers, as required.

Answer:

11

4 · Reviewdoes it hold up?

Put $a=3$ , $b=3$ back in: $-\frac{x^6}{y^3}\times\frac{y^4}{x^6}\times\frac{4y^4}{x^2}=-4x^{6-6-2}y^{-3+4+4}=-\frac{4y^5}{x^2}$ , which is the right side with $c=5$ .

Another way: Match exponents first: for $x$ , $2a-2b-2=-2$ ; for $y$ , $-a+4+4=c$ . In the range this allows $(a,b)=(2,2)$ or $(a,b)=(3,3)$ or $(a,b)=(4,4)$ . The sign then rules out $a=2$ and $a=4$ (even power, positive coefficient), leaving $a=3$ , $b=3$ , $c=5$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Expanding powers of quotients and collecting exponents of $x$ and $y$.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Deciding the sign of a power of a negative number.

💡Takeaway. Before you chase exponents, check the sign -- it can settle an unknown in one glance.

Variant 4 easy

In the equation $\left(-\frac{x^3}{y}\right)^a\times\left(\frac{y}{x^b}\right)^3\div\left(-\frac{x^3}{2y^2}\right)^2=-\frac{4y^c}{x^3}$ , the numbers $a$ , $b$ , and $c$ are natural numbers and $2\le a\le4$ . Find $a+b+c$ .

Your answer

Show solution

1 · Understandwhat's really being asked

An equation between monomial expressions contains unknown natural-number exponents $a$ , $b$ , $c$ with $2\le a\le4$ . Find $a+b+c$ .

Givens

$\left(-\frac{x^3}{y}\right)^a\times\left(\frac{y}{x^b}\right)^3\div\left(-\frac{x^3}{2y^2}\right)^2=-\frac{4y^c}{x^3}$
$a$ , $b$ , $c$ are natural numbers.
$2\le a\le4$

Unknowns

$a+b+c$

Constraints

Both sides must agree in sign of the coefficient and in the exponent of each letter.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#7 Identify Subproblems

The sign of the left side depends only on whether $a$ is even or odd, so settle $a$ first by matching signs. With $a$ known, simplify the left side and match the exponents of $x$ and $y$ .

3 · Execute6 carry out the plan

1Look at the sign first

#3 Eliminate Possibilities 7.NS.A.2

\left(\frac{y}{x^b}\right)^3

has no minus sign and

\left(-\frac{x^3}{2y^2}\right)^2

is an even power, so only

\left(-\frac{x^3}{y}\right)^a

can make the coefficient negative. The coefficient on the left is

(-1)^a\div\frac{1}{4}

.

(-1)^a\times4=-4 \Rightarrow (-1)^a=-1

A negative number to an odd power stays negative; to an even power it turns positive.

2Pin down $a$

#3 Eliminate Possibilities 7.NS.A.2

a

must be odd and

2\le a\le4

, which rules out

2

and

4

.

a=3

Only one odd number fits in the range.

3Expand each power

#7 Identify Subproblems 8.EE.A.1

Raise every factor inside each parenthesis to its power.

\left(-\frac{x^3}{y}\right)^3=-\frac{x^9}{y^3},\ \left(\frac{y}{x^b}\right)^3=\frac{y^3}{x^{3b}},\ \left(-\frac{x^3}{2y^2}\right)^2=\frac{x^6}{4y^4}

(x^3)^3=x^9

,

(2y^2)^2=4y^4

.

4Turn the division into multiplication

#13 Convert to Algebra 8.EE.A.1

Dividing by

\frac{x^6}{4y^4}

means multiplying by

\frac{4y^4}{x^6}

. Then collect powers of

x

and

y

.

-\frac{x^9}{y^3}\times\frac{y^3}{x^{3b}}\times\frac{4y^4}{x^6}=-\frac{4y^4}{x^{3b-3}}

For

x

:

9-3b-6=3-3b

. For

y

:

-3+3+4=4

.

5Match exponents

#13 Convert to Algebra 8.EE.A.1

Compare with

-\frac{4y^c}{x^3}

: the powers of

y

and of

x

must agree.

c=4,\quad 3b-3=3 \Rightarrow b=2

Same letter, same exponent on both sides.

6Add

#13 Convert to Algebra 8.EE.A.1

Add the three values.

a+b+c=3+2+4=9

All three are natural numbers, as required.

Answer:

9

4 · Reviewdoes it hold up?

Put $a=3$ , $b=2$ back in: $-\frac{x^9}{y^3}\times\frac{y^3}{x^6}\times\frac{4y^4}{x^6}=-4x^{9-6-6}y^{-3+3+4}=-\frac{4y^4}{x^3}$ , which is the right side with $c=4$ .

Another way: Match exponents first: for $x$ , $3a-3b-6=-3$ ; for $y$ , $-a+3+4=c$ . In the range this allows $(a,b)=(2,1)$ or $(a,b)=(3,2)$ or $(a,b)=(4,3)$ . The sign then rules out $a=2$ and $a=4$ (even power, positive coefficient), leaving $a=3$ , $b=2$ , $c=4$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Expanding powers of quotients and collecting exponents of $x$ and $y$.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Deciding the sign of a power of a negative number.

💡Takeaway. Before you chase exponents, check the sign -- it can settle an unknown in one glance.

Variant 5 medium

In the equation $\left(-\frac{x^2}{y}\right)^a\times\left(\frac{y}{x^b}\right)^2\div\left(-\frac{x}{3y^2}\right)^2=\frac{9y^c}{x^2}$ , the numbers $a$ , $b$ , and $c$ are natural numbers and $1\le a\le3$ . Find $a+b+c$ .

Your answer

Show solution

1 · Understandwhat's really being asked

An equation between monomial expressions contains unknown natural-number exponents $a$ , $b$ , $c$ with $1\le a\le3$ . Find $a+b+c$ .

Givens

$\left(-\frac{x^2}{y}\right)^a\times\left(\frac{y}{x^b}\right)^2\div\left(-\frac{x}{3y^2}\right)^2=\frac{9y^c}{x^2}$
$a$ , $b$ , $c$ are natural numbers.
$1\le a\le3$

Unknowns

$a+b+c$

Constraints

Both sides must agree in sign of the coefficient and in the exponent of each letter.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#7 Identify Subproblems

The sign of the left side depends only on whether $a$ is even or odd, so settle $a$ first by matching signs. With $a$ known, simplify the left side and match the exponents of $x$ and $y$ .

3 · Execute6 carry out the plan

1Look at the sign first

#3 Eliminate Possibilities 7.NS.A.2

\left(\frac{y}{x^b}\right)^2

has no minus sign and

\left(-\frac{x}{3y^2}\right)^2

is an even power, so only

\left(-\frac{x^2}{y}\right)^a

can make the coefficient negative. The coefficient on the left is

(-1)^a\div\frac{1}{9}

.

(-1)^a\times9=9 \Rightarrow (-1)^a=1

A negative number to an odd power stays negative; to an even power it turns positive.

2Pin down $a$

#3 Eliminate Possibilities 7.NS.A.2

a

must be even and

1\le a\le3

, which rules out

1

and

3

.

a=2

Only one even number fits in the range.

3Expand each power

#7 Identify Subproblems 8.EE.A.1

Raise every factor inside each parenthesis to its power.

\left(-\frac{x^2}{y}\right)^2=\frac{x^4}{y^2},\ \left(\frac{y}{x^b}\right)^2=\frac{y^2}{x^{2b}},\ \left(-\frac{x}{3y^2}\right)^2=\frac{x^2}{9y^4}

(x^2)^2=x^4

,

(3y^2)^2=9y^4

.

4Turn the division into multiplication

#13 Convert to Algebra 8.EE.A.1

Dividing by

\frac{x^2}{9y^4}

means multiplying by

\frac{9y^4}{x^2}

. Then collect powers of

x

and

y

.

\frac{x^4}{y^2}\times\frac{y^2}{x^{2b}}\times\frac{9y^4}{x^2}=\frac{9y^4}{x^{2b-2}}

For

x

:

4-2b-2=2-2b

. For

y

:

-2+2+4=4

.

5Match exponents

#13 Convert to Algebra 8.EE.A.1

Compare with

\frac{9y^c}{x^2}

: the powers of

y

and of

x

must agree.

c=4,\quad 2b-2=2 \Rightarrow b=2

Same letter, same exponent on both sides.

6Add

#13 Convert to Algebra 8.EE.A.1

Add the three values.

a+b+c=2+2+4=8

All three are natural numbers, as required.

Answer:

8

4 · Reviewdoes it hold up?

Put $a=2$ , $b=2$ back in: $\frac{x^4}{y^2}\times\frac{y^2}{x^4}\times\frac{9y^4}{x^2}=9x^{4-4-2}y^{-2+2+4}=\frac{9y^4}{x^2}$ , which is the right side with $c=4$ .

Another way: Match exponents first: for $x$ , $2a-2b-2=-2$ ; for $y$ , $-a+2+4=c$ . In the range this allows $(a,b)=(1,1)$ or $(a,b)=(2,2)$ or $(a,b)=(3,3)$ . The sign then rules out $a=1$ and $a=3$ (odd power, negative coefficient), leaving $a=2$ , $b=2$ , $c=4$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Expanding powers of quotients and collecting exponents of $x$ and $y$.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Deciding the sign of a power of a negative number.

💡Takeaway. Before you chase exponents, check the sign -- it can settle an unknown in one glance.

Variant 6 medium

In the equation $\left(-\frac{x^3}{y^2}\right)^a\times\left(\frac{y^2}{x^b}\right)^3\div\left(-\frac{x}{2y^2}\right)^2=-\frac{4y^c}{x^2}$ , the numbers $a$ , $b$ , and $c$ are natural numbers and $2\le a\le4$ . Find $a+b+c$ .

Your answer

Show solution

1 · Understandwhat's really being asked

An equation between monomial expressions contains unknown natural-number exponents $a$ , $b$ , $c$ with $2\le a\le4$ . Find $a+b+c$ .

Givens

$\left(-\frac{x^3}{y^2}\right)^a\times\left(\frac{y^2}{x^b}\right)^3\div\left(-\frac{x}{2y^2}\right)^2=-\frac{4y^c}{x^2}$
$a$ , $b$ , $c$ are natural numbers.
$2\le a\le4$

Unknowns

$a+b+c$

Constraints

Both sides must agree in sign of the coefficient and in the exponent of each letter.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#7 Identify Subproblems

The sign of the left side depends only on whether $a$ is even or odd, so settle $a$ first by matching signs. With $a$ known, simplify the left side and match the exponents of $x$ and $y$ .

3 · Execute6 carry out the plan

1Look at the sign first

#3 Eliminate Possibilities 7.NS.A.2

\left(\frac{y^2}{x^b}\right)^3

has no minus sign and

\left(-\frac{x}{2y^2}\right)^2

is an even power, so only

\left(-\frac{x^3}{y^2}\right)^a

can make the coefficient negative. The coefficient on the left is

(-1)^a\div\frac{1}{4}

.

(-1)^a\times4=-4 \Rightarrow (-1)^a=-1

A negative number to an odd power stays negative; to an even power it turns positive.

2Pin down $a$

#3 Eliminate Possibilities 7.NS.A.2

a

must be odd and

2\le a\le4

, which rules out

2

and

4

.

a=3

Only one odd number fits in the range.

3Expand each power

#7 Identify Subproblems 8.EE.A.1

Raise every factor inside each parenthesis to its power.

\left(-\frac{x^3}{y^2}\right)^3=-\frac{x^9}{y^6},\ \left(\frac{y^2}{x^b}\right)^3=\frac{y^6}{x^{3b}},\ \left(-\frac{x}{2y^2}\right)^2=\frac{x^2}{4y^4}

(x^3)^3=x^9

,

(y^2)^3=y^6

,

(2y^2)^2=4y^4

.

4Turn the division into multiplication

#13 Convert to Algebra 8.EE.A.1

Dividing by

\frac{x^2}{4y^4}

means multiplying by

\frac{4y^4}{x^2}

. Then collect powers of

x

and

y

.

-\frac{x^9}{y^6}\times\frac{y^6}{x^{3b}}\times\frac{4y^4}{x^2}=-\frac{4y^4}{x^{3b-7}}

For

x

:

9-3b-2=7-3b

. For

y

:

-6+6+4=4

.

5Match exponents

#13 Convert to Algebra 8.EE.A.1

Compare with

-\frac{4y^c}{x^2}

: the powers of

y

and of

x

must agree.

c=4,\quad 3b-7=2 \Rightarrow b=3

Same letter, same exponent on both sides.

6Add

#13 Convert to Algebra 8.EE.A.1

Add the three values.

a+b+c=3+3+4=10

All three are natural numbers, as required.

Answer:

10

4 · Reviewdoes it hold up?

Put $a=3$ , $b=3$ back in: $-\frac{x^9}{y^6}\times\frac{y^6}{x^9}\times\frac{4y^4}{x^2}=-4x^{9-9-2}y^{-6+6+4}=-\frac{4y^4}{x^2}$ , which is the right side with $c=4$ .

Another way: Match exponents first: for $x$ , $3a-3b-2=-2$ ; for $y$ , $-2a+6+4=c$ . In the range this allows $(a,b)=(2,2)$ or $(a,b)=(3,3)$ or $(a,b)=(4,4)$ . The sign then rules out $a=2$ and $a=4$ (even power, positive coefficient), leaving $a=3$ , $b=3$ , $c=4$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Expanding powers of quotients and collecting exponents of $x$ and $y$.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Deciding the sign of a power of a negative number.

💡Takeaway. Before you chase exponents, check the sign -- it can settle an unknown in one glance.

Variant 7 medium

In the equation $\left(-\frac{x^3}{y}\right)^a\times\left(\frac{y^3}{x^b}\right)^3\div\left(-\frac{x}{3y^2}\right)^2=\frac{9y^c}{x^2}$ , the numbers $a$ , $b$ , and $c$ are natural numbers and $1\le a\le3$ . Find $a+b+c$ .

Your answer

Show solution

1 · Understandwhat's really being asked

An equation between monomial expressions contains unknown natural-number exponents $a$ , $b$ , $c$ with $1\le a\le3$ . Find $a+b+c$ .

Givens

$\left(-\frac{x^3}{y}\right)^a\times\left(\frac{y^3}{x^b}\right)^3\div\left(-\frac{x}{3y^2}\right)^2=\frac{9y^c}{x^2}$
$a$ , $b$ , $c$ are natural numbers.
$1\le a\le3$

Unknowns

$a+b+c$

Constraints

Both sides must agree in sign of the coefficient and in the exponent of each letter.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#7 Identify Subproblems

The sign of the left side depends only on whether $a$ is even or odd, so settle $a$ first by matching signs. With $a$ known, simplify the left side and match the exponents of $x$ and $y$ .

3 · Execute6 carry out the plan

1Look at the sign first

#3 Eliminate Possibilities 7.NS.A.2

\left(\frac{y^3}{x^b}\right)^3

has no minus sign and

\left(-\frac{x}{3y^2}\right)^2

is an even power, so only

\left(-\frac{x^3}{y}\right)^a

can make the coefficient negative. The coefficient on the left is

(-1)^a\div\frac{1}{9}

.

(-1)^a\times9=9 \Rightarrow (-1)^a=1

A negative number to an odd power stays negative; to an even power it turns positive.

2Pin down $a$

#3 Eliminate Possibilities 7.NS.A.2

a

must be even and

1\le a\le3

, which rules out

1

and

3

.

a=2

Only one even number fits in the range.

3Expand each power

#7 Identify Subproblems 8.EE.A.1

Raise every factor inside each parenthesis to its power.

\left(-\frac{x^3}{y}\right)^2=\frac{x^6}{y^2},\ \left(\frac{y^3}{x^b}\right)^3=\frac{y^9}{x^{3b}},\ \left(-\frac{x}{3y^2}\right)^2=\frac{x^2}{9y^4}

(x^3)^2=x^6

,

(y^3)^3=y^9

,

(3y^2)^2=9y^4

.

4Turn the division into multiplication

#13 Convert to Algebra 8.EE.A.1

Dividing by

\frac{x^2}{9y^4}

means multiplying by

\frac{9y^4}{x^2}

. Then collect powers of

x

and

y

.

\frac{x^6}{y^2}\times\frac{y^9}{x^{3b}}\times\frac{9y^4}{x^2}=\frac{9y^{11}}{x^{3b-4}}

For

x

:

6-3b-2=4-3b

. For

y

:

-2+9+4=11

.

5Match exponents

#13 Convert to Algebra 8.EE.A.1

Compare with

\frac{9y^c}{x^2}

: the powers of

y

and of

x

must agree.

c=11,\quad 3b-4=2 \Rightarrow b=2

Same letter, same exponent on both sides.

6Add

#13 Convert to Algebra 8.EE.A.1

Add the three values.

a+b+c=2+2+11=15

All three are natural numbers, as required.

Answer:

15

4 · Reviewdoes it hold up?

Put $a=2$ , $b=2$ back in: $\frac{x^6}{y^2}\times\frac{y^9}{x^6}\times\frac{9y^4}{x^2}=9x^{6-6-2}y^{-2+9+4}=\frac{9y^{11}}{x^2}$ , which is the right side with $c=11$ .

Another way: Match exponents first: for $x$ , $3a-3b-2=-2$ ; for $y$ , $-a+9+4=c$ . In the range this allows $(a,b)=(1,1)$ or $(a,b)=(2,2)$ or $(a,b)=(3,3)$ . The sign then rules out $a=1$ and $a=3$ (odd power, negative coefficient), leaving $a=2$ , $b=2$ , $c=11$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Expanding powers of quotients and collecting exponents of $x$ and $y$.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Deciding the sign of a power of a negative number.

💡Takeaway. Before you chase exponents, check the sign -- it can settle an unknown in one glance.

Variant 8 medium

In the equation $\left(-\frac{x^2}{y^2}\right)^a\times\left(\frac{y^2}{x^b}\right)^2\div\left(-\frac{x^2}{2y^2}\right)^2=\frac{4y^c}{x^2}$ , the numbers $a$ , $b$ , and $c$ are natural numbers and $1\le a\le3$ . Find $a+b+c$ .

Your answer

Show solution

1 · Understandwhat's really being asked

An equation between monomial expressions contains unknown natural-number exponents $a$ , $b$ , $c$ with $1\le a\le3$ . Find $a+b+c$ .

Givens

$\left(-\frac{x^2}{y^2}\right)^a\times\left(\frac{y^2}{x^b}\right)^2\div\left(-\frac{x^2}{2y^2}\right)^2=\frac{4y^c}{x^2}$
$a$ , $b$ , $c$ are natural numbers.
$1\le a\le3$

Unknowns

$a+b+c$

Constraints

Both sides must agree in sign of the coefficient and in the exponent of each letter.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#7 Identify Subproblems

The sign of the left side depends only on whether $a$ is even or odd, so settle $a$ first by matching signs. With $a$ known, simplify the left side and match the exponents of $x$ and $y$ .

3 · Execute6 carry out the plan

1Look at the sign first

#3 Eliminate Possibilities 7.NS.A.2

\left(\frac{y^2}{x^b}\right)^2

has no minus sign and

\left(-\frac{x^2}{2y^2}\right)^2

is an even power, so only

\left(-\frac{x^2}{y^2}\right)^a

can make the coefficient negative. The coefficient on the left is

(-1)^a\div\frac{1}{4}

.

(-1)^a\times4=4 \Rightarrow (-1)^a=1

A negative number to an odd power stays negative; to an even power it turns positive.

2Pin down $a$

#3 Eliminate Possibilities 7.NS.A.2

a

must be even and

1\le a\le3

, which rules out

1

and

3

.

a=2

Only one even number fits in the range.

3Expand each power

#7 Identify Subproblems 8.EE.A.1

Raise every factor inside each parenthesis to its power.

\left(-\frac{x^2}{y^2}\right)^2=\frac{x^4}{y^4},\ \left(\frac{y^2}{x^b}\right)^2=\frac{y^4}{x^{2b}},\ \left(-\frac{x^2}{2y^2}\right)^2=\frac{x^4}{4y^4}

(x^2)^2=x^4

,

(y^2)^2=y^4

,

(2y^2)^2=4y^4

.

4Turn the division into multiplication

#13 Convert to Algebra 8.EE.A.1

Dividing by

\frac{x^4}{4y^4}

means multiplying by

\frac{4y^4}{x^4}

. Then collect powers of

x

and

y

.

\frac{x^4}{y^4}\times\frac{y^4}{x^{2b}}\times\frac{4y^4}{x^4}=\frac{4y^4}{x^{2b}}

For

x

:

4-2b-4=-2b

. For

y

:

-4+4+4=4

.

5Match exponents

#13 Convert to Algebra 8.EE.A.1

Compare with

\frac{4y^c}{x^2}

: the powers of

y

and of

x

must agree.

c=4,\quad 2b=2 \Rightarrow b=1

Same letter, same exponent on both sides.

6Add

#13 Convert to Algebra 8.EE.A.1

Add the three values.

a+b+c=2+1+4=7

All three are natural numbers, as required.

Answer:

7

4 · Reviewdoes it hold up?

Put $a=2$ , $b=1$ back in: $\frac{x^4}{y^4}\times\frac{y^4}{x^2}\times\frac{4y^4}{x^4}=4x^{4-2-4}y^{-4+4+4}=\frac{4y^4}{x^2}$ , which is the right side with $c=4$ .

Another way: Match exponents first: for $x$ , $2a-2b-4=-2$ ; for $y$ , $-2a+4+4=c$ . In the range this allows $(a,b)=(2,1)$ or $(a,b)=(3,2)$ . The sign then rules out $a=3$ (odd power, negative coefficient), leaving $a=2$ , $b=1$ , $c=4$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Expanding powers of quotients and collecting exponents of $x$ and $y$.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Deciding the sign of a power of a negative number.

💡Takeaway. Before you chase exponents, check the sign -- it can settle an unknown in one glance.

Variant 9 hard

In the equation $\left(-\frac{x^2}{y}\right)^a\times\left(\frac{y^2}{x^b}\right)^2\div\left(-\frac{x^2}{3y^2}\right)^2=-\frac{9y^c}{x^2}$ , the numbers $a$ , $b$ , and $c$ are natural numbers and $2\le a\le4$ . Find $a+b+c$ .

Your answer

Show solution

1 · Understandwhat's really being asked

An equation between monomial expressions contains unknown natural-number exponents $a$ , $b$ , $c$ with $2\le a\le4$ . Find $a+b+c$ .

Givens

$\left(-\frac{x^2}{y}\right)^a\times\left(\frac{y^2}{x^b}\right)^2\div\left(-\frac{x^2}{3y^2}\right)^2=-\frac{9y^c}{x^2}$
$a$ , $b$ , $c$ are natural numbers.
$2\le a\le4$

Unknowns

$a+b+c$

Constraints

Both sides must agree in sign of the coefficient and in the exponent of each letter.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#7 Identify Subproblems

The sign of the left side depends only on whether $a$ is even or odd, so settle $a$ first by matching signs. With $a$ known, simplify the left side and match the exponents of $x$ and $y$ .

3 · Execute6 carry out the plan

1Look at the sign first

#3 Eliminate Possibilities 7.NS.A.2

\left(\frac{y^2}{x^b}\right)^2

has no minus sign and

\left(-\frac{x^2}{3y^2}\right)^2

is an even power, so only

\left(-\frac{x^2}{y}\right)^a

can make the coefficient negative. The coefficient on the left is

(-1)^a\div\frac{1}{9}

.

(-1)^a\times9=-9 \Rightarrow (-1)^a=-1

A negative number to an odd power stays negative; to an even power it turns positive.

2Pin down $a$

#3 Eliminate Possibilities 7.NS.A.2

a

must be odd and

2\le a\le4

, which rules out

2

and

4

.

a=3

Only one odd number fits in the range.

3Expand each power

#7 Identify Subproblems 8.EE.A.1

Raise every factor inside each parenthesis to its power.

\left(-\frac{x^2}{y}\right)^3=-\frac{x^6}{y^3},\ \left(\frac{y^2}{x^b}\right)^2=\frac{y^4}{x^{2b}},\ \left(-\frac{x^2}{3y^2}\right)^2=\frac{x^4}{9y^4}

(x^2)^3=x^6

,

(y^2)^2=y^4

,

(3y^2)^2=9y^4

.

4Turn the division into multiplication

#13 Convert to Algebra 8.EE.A.1

Dividing by

\frac{x^4}{9y^4}

means multiplying by

\frac{9y^4}{x^4}

. Then collect powers of

x

and

y

.

-\frac{x^6}{y^3}\times\frac{y^4}{x^{2b}}\times\frac{9y^4}{x^4}=-\frac{9y^5}{x^{2b-2}}

For

x

:

6-2b-4=2-2b

. For

y

:

-3+4+4=5

.

5Match exponents

#13 Convert to Algebra 8.EE.A.1

Compare with

-\frac{9y^c}{x^2}

: the powers of

y

and of

x

must agree.

c=5,\quad 2b-2=2 \Rightarrow b=2

Same letter, same exponent on both sides.

6Add

#13 Convert to Algebra 8.EE.A.1

Add the three values.

a+b+c=3+2+5=10

All three are natural numbers, as required.

Answer:

10

4 · Reviewdoes it hold up?

Put $a=3$ , $b=2$ back in: $-\frac{x^6}{y^3}\times\frac{y^4}{x^4}\times\frac{9y^4}{x^4}=-9x^{6-4-4}y^{-3+4+4}=-\frac{9y^5}{x^2}$ , which is the right side with $c=5$ .

Another way: Match exponents first: for $x$ , $2a-2b-4=-2$ ; for $y$ , $-a+4+4=c$ . In the range this allows $(a,b)=(2,1)$ or $(a,b)=(3,2)$ or $(a,b)=(4,3)$ . The sign then rules out $a=2$ and $a=4$ (even power, positive coefficient), leaving $a=3$ , $b=2$ , $c=5$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Expanding powers of quotients and collecting exponents of $x$ and $y$.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Deciding the sign of a power of a negative number.

💡Takeaway. Before you chase exponents, check the sign -- it can settle an unknown in one glance.

Variant 10 hard

In the equation $\left(-\frac{x^3}{y^3}\right)^a\times\left(\frac{y^2}{x^b}\right)^3\div\left(-\frac{x}{3y^2}\right)^2=\frac{9y^c}{x^2}$ , the numbers $a$ , $b$ , and $c$ are natural numbers and $1\le a\le3$ . Find $a+b+c$ .

Your answer

Show solution

1 · Understandwhat's really being asked

An equation between monomial expressions contains unknown natural-number exponents $a$ , $b$ , $c$ with $1\le a\le3$ . Find $a+b+c$ .

Givens

$\left(-\frac{x^3}{y^3}\right)^a\times\left(\frac{y^2}{x^b}\right)^3\div\left(-\frac{x}{3y^2}\right)^2=\frac{9y^c}{x^2}$
$a$ , $b$ , $c$ are natural numbers.
$1\le a\le3$

Unknowns

$a+b+c$

Constraints

Both sides must agree in sign of the coefficient and in the exponent of each letter.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#7 Identify Subproblems

The sign of the left side depends only on whether $a$ is even or odd, so settle $a$ first by matching signs. With $a$ known, simplify the left side and match the exponents of $x$ and $y$ .

3 · Execute6 carry out the plan

1Look at the sign first

#3 Eliminate Possibilities 7.NS.A.2

\left(\frac{y^2}{x^b}\right)^3

has no minus sign and

\left(-\frac{x}{3y^2}\right)^2

is an even power, so only

\left(-\frac{x^3}{y^3}\right)^a

can make the coefficient negative. The coefficient on the left is

(-1)^a\div\frac{1}{9}

.

(-1)^a\times9=9 \Rightarrow (-1)^a=1

A negative number to an odd power stays negative; to an even power it turns positive.

2Pin down $a$

#3 Eliminate Possibilities 7.NS.A.2

a

must be even and

1\le a\le3

, which rules out

1

and

3

.

a=2

Only one even number fits in the range.

3Expand each power

#7 Identify Subproblems 8.EE.A.1

Raise every factor inside each parenthesis to its power.

\left(-\frac{x^3}{y^3}\right)^2=\frac{x^6}{y^6},\ \left(\frac{y^2}{x^b}\right)^3=\frac{y^6}{x^{3b}},\ \left(-\frac{x}{3y^2}\right)^2=\frac{x^2}{9y^4}

(x^3)^2=x^6

,

(y^2)^3=y^6

,

(3y^2)^2=9y^4

.

4Turn the division into multiplication

#13 Convert to Algebra 8.EE.A.1

Dividing by

\frac{x^2}{9y^4}

means multiplying by

\frac{9y^4}{x^2}

. Then collect powers of

x

and

y

.

\frac{x^6}{y^6}\times\frac{y^6}{x^{3b}}\times\frac{9y^4}{x^2}=\frac{9y^4}{x^{3b-4}}

For

x

:

6-3b-2=4-3b

. For

y

:

-6+6+4=4

.

5Match exponents

#13 Convert to Algebra 8.EE.A.1

Compare with

\frac{9y^c}{x^2}

: the powers of

y

and of

x

must agree.

c=4,\quad 3b-4=2 \Rightarrow b=2

Same letter, same exponent on both sides.

6Add

#13 Convert to Algebra 8.EE.A.1

Add the three values.

a+b+c=2+2+4=8

All three are natural numbers, as required.

Answer:

8

4 · Reviewdoes it hold up?

Put $a=2$ , $b=2$ back in: $\frac{x^6}{y^6}\times\frac{y^6}{x^6}\times\frac{9y^4}{x^2}=9x^{6-6-2}y^{-6+6+4}=\frac{9y^4}{x^2}$ , which is the right side with $c=4$ .

Another way: Match exponents first: for $x$ , $3a-3b-2=-2$ ; for $y$ , $-3a+6+4=c$ . In the range this allows $(a,b)=(1,1)$ or $(a,b)=(2,2)$ or $(a,b)=(3,3)$ . The sign then rules out $a=1$ and $a=3$ (odd power, negative coefficient), leaving $a=2$ , $b=2$ , $c=4$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Expanding powers of quotients and collecting exponents of $x$ and $y$.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Deciding the sign of a power of a negative number.

💡Takeaway. Before you chase exponents, check the sign -- it can settle an unknown in one glance.

Variant 11 hard

In the equation $\left(-\frac{x^4}{y}\right)^a\times\left(\frac{y^2}{x^b}\right)^2\div\left(-\frac{x^3}{3y^2}\right)^2=\frac{9y^c}{x^2}$ , the numbers $a$ , $b$ , and $c$ are natural numbers and $1\le a\le3$ . Find $a+b+c$ .

Your answer

Show solution

1 · Understandwhat's really being asked

An equation between monomial expressions contains unknown natural-number exponents $a$ , $b$ , $c$ with $1\le a\le3$ . Find $a+b+c$ .

Givens

$\left(-\frac{x^4}{y}\right)^a\times\left(\frac{y^2}{x^b}\right)^2\div\left(-\frac{x^3}{3y^2}\right)^2=\frac{9y^c}{x^2}$
$a$ , $b$ , $c$ are natural numbers.
$1\le a\le3$

Unknowns

$a+b+c$

Constraints

Both sides must agree in sign of the coefficient and in the exponent of each letter.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#7 Identify Subproblems

The sign of the left side depends only on whether $a$ is even or odd, so settle $a$ first by matching signs. With $a$ known, simplify the left side and match the exponents of $x$ and $y$ .

3 · Execute6 carry out the plan

1Look at the sign first

#3 Eliminate Possibilities 7.NS.A.2

\left(\frac{y^2}{x^b}\right)^2

has no minus sign and

\left(-\frac{x^3}{3y^2}\right)^2

is an even power, so only

\left(-\frac{x^4}{y}\right)^a

can make the coefficient negative. The coefficient on the left is

(-1)^a\div\frac{1}{9}

.

(-1)^a\times9=9 \Rightarrow (-1)^a=1

A negative number to an odd power stays negative; to an even power it turns positive.

2Pin down $a$

#3 Eliminate Possibilities 7.NS.A.2

a

must be even and

1\le a\le3

, which rules out

1

and

3

.

a=2

Only one even number fits in the range.

3Expand each power

#7 Identify Subproblems 8.EE.A.1

Raise every factor inside each parenthesis to its power.

\left(-\frac{x^4}{y}\right)^2=\frac{x^8}{y^2},\ \left(\frac{y^2}{x^b}\right)^2=\frac{y^4}{x^{2b}},\ \left(-\frac{x^3}{3y^2}\right)^2=\frac{x^6}{9y^4}

(x^4)^2=x^8

,

(y^2)^2=y^4

,

(3y^2)^2=9y^4

.

4Turn the division into multiplication

#13 Convert to Algebra 8.EE.A.1

Dividing by

\frac{x^6}{9y^4}

means multiplying by

\frac{9y^4}{x^6}

. Then collect powers of

x

and

y

.

\frac{x^8}{y^2}\times\frac{y^4}{x^{2b}}\times\frac{9y^4}{x^6}=\frac{9y^6}{x^{2b-2}}

For

x

:

8-2b-6=2-2b

. For

y

:

-2+4+4=6

.

5Match exponents

#13 Convert to Algebra 8.EE.A.1

Compare with

\frac{9y^c}{x^2}

: the powers of

y

and of

x

must agree.

c=6,\quad 2b-2=2 \Rightarrow b=2

Same letter, same exponent on both sides.

6Add

#13 Convert to Algebra 8.EE.A.1

Add the three values.

a+b+c=2+2+6=10

All three are natural numbers, as required.

Answer:

10

4 · Reviewdoes it hold up?

Put $a=2$ , $b=2$ back in: $\frac{x^8}{y^2}\times\frac{y^4}{x^4}\times\frac{9y^4}{x^6}=9x^{8-4-6}y^{-2+4+4}=\frac{9y^6}{x^2}$ , which is the right side with $c=6$ .

Another way: Match exponents first: for $x$ , $4a-2b-6=-2$ ; for $y$ , $-a+4+4=c$ . In the range this allows $(a,b)=(2,2)$ or $(a,b)=(3,4)$ . The sign then rules out $a=3$ (odd power, negative coefficient), leaving $a=2$ , $b=2$ , $c=6$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Expanding powers of quotients and collecting exponents of $x$ and $y$.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Deciding the sign of a power of a negative number.

💡Takeaway. Before you chase exponents, check the sign -- it can settle an unknown in one glance.

Variant 12 hard

In the equation $\left(-\frac{x^2}{y^3}\right)^a\times\left(\frac{y^3}{x^b}\right)^2\div\left(-\frac{x^2}{3y^2}\right)^2=\frac{9y^c}{x^2}$ , the numbers $a$ , $b$ , and $c$ are natural numbers and $1\le a\le3$ . Find $a+b+c$ .

Your answer

Show solution

1 · Understandwhat's really being asked

An equation between monomial expressions contains unknown natural-number exponents $a$ , $b$ , $c$ with $1\le a\le3$ . Find $a+b+c$ .

Givens

$\left(-\frac{x^2}{y^3}\right)^a\times\left(\frac{y^3}{x^b}\right)^2\div\left(-\frac{x^2}{3y^2}\right)^2=\frac{9y^c}{x^2}$
$a$ , $b$ , $c$ are natural numbers.
$1\le a\le3$

Unknowns

$a+b+c$

Constraints

Both sides must agree in sign of the coefficient and in the exponent of each letter.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #3 Eliminate Possibilities#7 Identify Subproblems

The sign of the left side depends only on whether $a$ is even or odd, so settle $a$ first by matching signs. With $a$ known, simplify the left side and match the exponents of $x$ and $y$ .

3 · Execute6 carry out the plan

1Look at the sign first

#3 Eliminate Possibilities 7.NS.A.2

\left(\frac{y^3}{x^b}\right)^2

has no minus sign and

\left(-\frac{x^2}{3y^2}\right)^2

is an even power, so only

\left(-\frac{x^2}{y^3}\right)^a

can make the coefficient negative. The coefficient on the left is

(-1)^a\div\frac{1}{9}

.

(-1)^a\times9=9 \Rightarrow (-1)^a=1

A negative number to an odd power stays negative; to an even power it turns positive.

2Pin down $a$

#3 Eliminate Possibilities 7.NS.A.2

a

must be even and

1\le a\le3

, which rules out

1

and

3

.

a=2

Only one even number fits in the range.

3Expand each power

#7 Identify Subproblems 8.EE.A.1

Raise every factor inside each parenthesis to its power.

\left(-\frac{x^2}{y^3}\right)^2=\frac{x^4}{y^6},\ \left(\frac{y^3}{x^b}\right)^2=\frac{y^6}{x^{2b}},\ \left(-\frac{x^2}{3y^2}\right)^2=\frac{x^4}{9y^4}

(x^2)^2=x^4

,

(y^3)^2=y^6

,

(3y^2)^2=9y^4

.

4Turn the division into multiplication

#13 Convert to Algebra 8.EE.A.1

Dividing by

\frac{x^4}{9y^4}

means multiplying by

\frac{9y^4}{x^4}

. Then collect powers of

x

and

y

.

\frac{x^4}{y^6}\times\frac{y^6}{x^{2b}}\times\frac{9y^4}{x^4}=\frac{9y^4}{x^{2b}}

For

x

:

4-2b-4=-2b

. For

y

:

-6+6+4=4

.

5Match exponents

#13 Convert to Algebra 8.EE.A.1

Compare with

\frac{9y^c}{x^2}

: the powers of

y

and of

x

must agree.

c=4,\quad 2b=2 \Rightarrow b=1

Same letter, same exponent on both sides.

6Add

#13 Convert to Algebra 8.EE.A.1

Add the three values.

a+b+c=2+1+4=7

All three are natural numbers, as required.

Answer:

7

4 · Reviewdoes it hold up?

Put $a=2$ , $b=1$ back in: $\frac{x^4}{y^6}\times\frac{y^6}{x^2}\times\frac{9y^4}{x^4}=9x^{4-2-4}y^{-6+6+4}=\frac{9y^4}{x^2}$ , which is the right side with $c=4$ .

Another way: Match exponents first: for $x$ , $2a-2b-4=-2$ ; for $y$ , $-3a+6+4=c$ . In the range this allows $(a,b)=(2,1)$ or $(a,b)=(3,2)$ . The sign then rules out $a=3$ (odd power, negative coefficient), leaving $a=2$ , $b=1$ , $c=4$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Expanding powers of quotients and collecting exponents of $x$ and $y$.
7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Deciding the sign of a power of a negative number.

💡Takeaway. Before you chase exponents, check the sign -- it can settle an unknown in one glance.

Let the sign decide the exponent's parity · 12 practice problems

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