Practice · Pair 2s with 5s to make powers of ten · Sensim Math

Variant 1 easy

The number $2^{10}\times5^{14}$ is a natural number with $n$ digits. Find the value of $n$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Find how many digits the natural number $2^{10}\times5^{14}$ has.

Givens

The number is $2^{10}\times5^{14}$ .

Unknowns

The number of digits $n$ .

Constraints

Multiplying by $10$ adds one zero, so a number written as (a number) $\times10^k$ is easy to count.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Each $2\times5$ makes a $10$ . Pair up as many $2$ s with $5$ s as possible, leaving a small leftover number times a power of $10$ .

3 · Execute4 carry out the plan

1Split off the extra $5$ s

#7 Identify Subproblems 8.EE.A.1

There are

10

twos and

14

fives, so set aside

4

fives.

5^{14}=5^{10}\times5^4

10+4=14

.

2Pair twos with fives

#15 Organize Information in More Ways 8.EE.A.1

Powers with the same exponent multiply into one power of the product.

2^{10}\times5^{10}=(2\times5)^{10}=10^{10}

Each pair is a factor of

10

.

3Write as a number times a power of $10$

#15 Organize Information in More Ways 8.EE.A.3

The leftover is

5^4=625

.

2^{10}\times5^{14}=625\times10^{10}

5^4=625

.

4Count the digits

#9 Solve an Easier Related Problem 8.EE.A.3

Multiplying

625

by

10^{10}

writes

10

zeros after it:

625\times10^2=62500

has

3+2

digits, and in the same way this number has

3+10

digits.

n=3+10=13

Each factor of

10

adds exactly one digit.

Answer:

13

4 · Reviewdoes it hold up?

In scientific notation the number is $6.25\times10^{12}$ , which lies between $10^{12}$ (the smallest $13$ -digit number) and $10^{13}$ . So it has $13$ digits.

Another way: Pair all of it at once using scientific notation: $2^{10}\times5^{14}=6.25\times10^{12}$ , and a number $c\times10^{k}$ with $1\le c<10$ has $k+1$ digits, giving $13$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Splitting $5^{14}$ and combining $2^{10}\times5^{10}$ into $10^{10}$.
8.EE.A.3 Use numbers expressed in the form of a single digit times a power of 10 — Reading the size of a number written with a power of $10$.

💡Takeaway. Every

2

paired with a

5

is a

10

-- count the tens and you've counted the zeros.

Variant 2 easy

The number $2^{14}\times5^8$ is a natural number with $n$ digits. Find the value of $n$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Find how many digits the natural number $2^{14}\times5^8$ has.

Givens

The number is $2^{14}\times5^8$ .

Unknowns

The number of digits $n$ .

Constraints

Multiplying by $10$ adds one zero, so a number written as (a number) $\times10^k$ is easy to count.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Each $2\times5$ makes a $10$ . Pair up as many $2$ s with $5$ s as possible, leaving a small leftover number times a power of $10$ .

3 · Execute4 carry out the plan

1Split off the extra $2$ s

#7 Identify Subproblems 8.EE.A.1

There are

14

twos and

8

fives, so set aside

6

twos.

2^{14}=2^8\times2^6

8+6=14

.

2Pair twos with fives

#15 Organize Information in More Ways 8.EE.A.1

Powers with the same exponent multiply into one power of the product.

2^8\times5^8=(2\times5)^{8}=10^8

Each pair is a factor of

10

.

3Write as a number times a power of $10$

#15 Organize Information in More Ways 8.EE.A.3

The leftover is

2^6=64

.

2^{14}\times5^8=64\times10^8

2^6=64

.

4Count the digits

#9 Solve an Easier Related Problem 8.EE.A.3

Multiplying

64

by

10^8

writes

8

zeros after it:

64\times10^2=6400

has

2+2

digits, and in the same way this number has

2+8

digits.

n=2+8=10

Each factor of

10

adds exactly one digit.

Answer:

10

4 · Reviewdoes it hold up?

In scientific notation the number is $6.4\times10^9$ , which lies between $10^9$ (the smallest $10$ -digit number) and $10^{10}$ . So it has $10$ digits.

Another way: Pair all of it at once using scientific notation: $2^{14}\times5^8=6.4\times10^9$ , and a number $c\times10^{k}$ with $1\le c<10$ has $k+1$ digits, giving $10$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Splitting $2^{14}$ and combining $2^8\times5^8$ into $10^8$.
8.EE.A.3 Use numbers expressed in the form of a single digit times a power of 10 — Reading the size of a number written with a power of $10$.

💡Takeaway. Every

2

paired with a

5

is a

10

-- count the tens and you've counted the zeros.

Variant 3 easy

The number $2^{16}\times5^{11}$ is a natural number with $n$ digits. Find the value of $n$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Find how many digits the natural number $2^{16}\times5^{11}$ has.

Givens

The number is $2^{16}\times5^{11}$ .

Unknowns

The number of digits $n$ .

Constraints

Multiplying by $10$ adds one zero, so a number written as (a number) $\times10^k$ is easy to count.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Each $2\times5$ makes a $10$ . Pair up as many $2$ s with $5$ s as possible, leaving a small leftover number times a power of $10$ .

3 · Execute4 carry out the plan

1Split off the extra $2$ s

#7 Identify Subproblems 8.EE.A.1

There are

16

twos and

11

fives, so set aside

5

twos.

2^{16}=2^{11}\times2^5

11+5=16

.

2Pair twos with fives

#15 Organize Information in More Ways 8.EE.A.1

Powers with the same exponent multiply into one power of the product.

2^{11}\times5^{11}=(2\times5)^{11}=10^{11}

Each pair is a factor of

10

.

3Write as a number times a power of $10$

#15 Organize Information in More Ways 8.EE.A.3

The leftover is

2^5=32

.

2^{16}\times5^{11}=32\times10^{11}

2^5=32

.

4Count the digits

#9 Solve an Easier Related Problem 8.EE.A.3

Multiplying

32

by

10^{11}

writes

11

zeros after it:

32\times10^2=3200

has

2+2

digits, and in the same way this number has

2+11

digits.

n=2+11=13

Each factor of

10

adds exactly one digit.

Answer:

13

4 · Reviewdoes it hold up?

In scientific notation the number is $3.2\times10^{12}$ , which lies between $10^{12}$ (the smallest $13$ -digit number) and $10^{13}$ . So it has $13$ digits.

Another way: Pair all of it at once using scientific notation: $2^{16}\times5^{11}=3.2\times10^{12}$ , and a number $c\times10^{k}$ with $1\le c<10$ has $k+1$ digits, giving $13$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Splitting $2^{16}$ and combining $2^{11}\times5^{11}$ into $10^{11}$.
8.EE.A.3 Use numbers expressed in the form of a single digit times a power of 10 — Reading the size of a number written with a power of $10$.

💡Takeaway. Every

2

paired with a

5

is a

10

-- count the tens and you've counted the zeros.

Variant 4 easy

The number $2^{12}\times5^{16}$ is a natural number with $n$ digits. Find the value of $n$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Find how many digits the natural number $2^{12}\times5^{16}$ has.

Givens

The number is $2^{12}\times5^{16}$ .

Unknowns

The number of digits $n$ .

Constraints

Multiplying by $10$ adds one zero, so a number written as (a number) $\times10^k$ is easy to count.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Each $2\times5$ makes a $10$ . Pair up as many $2$ s with $5$ s as possible, leaving a small leftover number times a power of $10$ .

3 · Execute4 carry out the plan

1Split off the extra $5$ s

#7 Identify Subproblems 8.EE.A.1

There are

12

twos and

16

fives, so set aside

4

fives.

5^{16}=5^{12}\times5^4

12+4=16

.

2Pair twos with fives

#15 Organize Information in More Ways 8.EE.A.1

Powers with the same exponent multiply into one power of the product.

2^{12}\times5^{12}=(2\times5)^{12}=10^{12}

Each pair is a factor of

10

.

3Write as a number times a power of $10$

#15 Organize Information in More Ways 8.EE.A.3

The leftover is

5^4=625

.

2^{12}\times5^{16}=625\times10^{12}

5^4=625

.

4Count the digits

#9 Solve an Easier Related Problem 8.EE.A.3

Multiplying

625

by

10^{12}

writes

12

zeros after it:

625\times10^2=62500

has

3+2

digits, and in the same way this number has

3+12

digits.

n=3+12=15

Each factor of

10

adds exactly one digit.

Answer:

15

4 · Reviewdoes it hold up?

In scientific notation the number is $6.25\times10^{14}$ , which lies between $10^{14}$ (the smallest $15$ -digit number) and $10^{15}$ . So it has $15$ digits.

Another way: Pair all of it at once using scientific notation: $2^{12}\times5^{16}=6.25\times10^{14}$ , and a number $c\times10^{k}$ with $1\le c<10$ has $k+1$ digits, giving $15$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Splitting $5^{16}$ and combining $2^{12}\times5^{12}$ into $10^{12}$.
8.EE.A.3 Use numbers expressed in the form of a single digit times a power of 10 — Reading the size of a number written with a power of $10$.

💡Takeaway. Every

2

paired with a

5

is a

10

-- count the tens and you've counted the zeros.

Variant 5 medium

The number $2^{15}\times5^{17}$ is a natural number with $n$ digits. Find the value of $n$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Find how many digits the natural number $2^{15}\times5^{17}$ has.

Givens

The number is $2^{15}\times5^{17}$ .

Unknowns

The number of digits $n$ .

Constraints

Multiplying by $10$ adds one zero, so a number written as (a number) $\times10^k$ is easy to count.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Each $2\times5$ makes a $10$ . Pair up as many $2$ s with $5$ s as possible, leaving a small leftover number times a power of $10$ .

3 · Execute4 carry out the plan

1Split off the extra $5$ s

#7 Identify Subproblems 8.EE.A.1

There are

15

twos and

17

fives, so set aside

2

fives.

5^{17}=5^{15}\times5^2

15+2=17

.

2Pair twos with fives

#15 Organize Information in More Ways 8.EE.A.1

Powers with the same exponent multiply into one power of the product.

2^{15}\times5^{15}=(2\times5)^{15}=10^{15}

Each pair is a factor of

10

.

3Write as a number times a power of $10$

#15 Organize Information in More Ways 8.EE.A.3

The leftover is

5^2=25

.

2^{15}\times5^{17}=25\times10^{15}

5^2=25

.

4Count the digits

#9 Solve an Easier Related Problem 8.EE.A.3

Multiplying

25

by

10^{15}

writes

15

zeros after it:

25\times10^2=2500

has

2+2

digits, and in the same way this number has

2+15

digits.

n=2+15=17

Each factor of

10

adds exactly one digit.

Answer:

17

4 · Reviewdoes it hold up?

In scientific notation the number is $2.5\times10^{16}$ , which lies between $10^{16}$ (the smallest $17$ -digit number) and $10^{17}$ . So it has $17$ digits.

Another way: Pair all of it at once using scientific notation: $2^{15}\times5^{17}=2.5\times10^{16}$ , and a number $c\times10^{k}$ with $1\le c<10$ has $k+1$ digits, giving $17$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Splitting $5^{17}$ and combining $2^{15}\times5^{15}$ into $10^{15}$.
8.EE.A.3 Use numbers expressed in the form of a single digit times a power of 10 — Reading the size of a number written with a power of $10$.

💡Takeaway. Every

2

paired with a

5

is a

10

-- count the tens and you've counted the zeros.

Variant 6 medium

The number $2^{13}\times5^{18}$ is a natural number with $n$ digits. Find the value of $n$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Find how many digits the natural number $2^{13}\times5^{18}$ has.

Givens

The number is $2^{13}\times5^{18}$ .

Unknowns

The number of digits $n$ .

Constraints

Multiplying by $10$ adds one zero, so a number written as (a number) $\times10^k$ is easy to count.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Each $2\times5$ makes a $10$ . Pair up as many $2$ s with $5$ s as possible, leaving a small leftover number times a power of $10$ .

3 · Execute4 carry out the plan

1Split off the extra $5$ s

#7 Identify Subproblems 8.EE.A.1

There are

13

twos and

18

fives, so set aside

5

fives.

5^{18}=5^{13}\times5^5

13+5=18

.

2Pair twos with fives

#15 Organize Information in More Ways 8.EE.A.1

Powers with the same exponent multiply into one power of the product.

2^{13}\times5^{13}=(2\times5)^{13}=10^{13}

Each pair is a factor of

10

.

3Write as a number times a power of $10$

#15 Organize Information in More Ways 8.EE.A.3

The leftover is

5^5=3125

.

2^{13}\times5^{18}=3125\times10^{13}

5^5=3125

.

4Count the digits

#9 Solve an Easier Related Problem 8.EE.A.3

Multiplying

3125

by

10^{13}

writes

13

zeros after it:

3125\times10^2=312500

has

4+2

digits, and in the same way this number has

4+13

digits.

n=4+13=17

Each factor of

10

adds exactly one digit.

Answer:

17

4 · Reviewdoes it hold up?

In scientific notation the number is $3.125\times10^{16}$ , which lies between $10^{16}$ (the smallest $17$ -digit number) and $10^{17}$ . So it has $17$ digits.

Another way: Pair all of it at once using scientific notation: $2^{13}\times5^{18}=3.125\times10^{16}$ , and a number $c\times10^{k}$ with $1\le c<10$ has $k+1$ digits, giving $17$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Splitting $5^{18}$ and combining $2^{13}\times5^{13}$ into $10^{13}$.
8.EE.A.3 Use numbers expressed in the form of a single digit times a power of 10 — Reading the size of a number written with a power of $10$.

💡Takeaway. Every

2

paired with a

5

is a

10

-- count the tens and you've counted the zeros.

Variant 7 medium

The number $2^{19}\times5^{22}$ is a natural number with $n$ digits. Find the value of $n$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Find how many digits the natural number $2^{19}\times5^{22}$ has.

Givens

The number is $2^{19}\times5^{22}$ .

Unknowns

The number of digits $n$ .

Constraints

Multiplying by $10$ adds one zero, so a number written as (a number) $\times10^k$ is easy to count.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Each $2\times5$ makes a $10$ . Pair up as many $2$ s with $5$ s as possible, leaving a small leftover number times a power of $10$ .

3 · Execute4 carry out the plan

1Split off the extra $5$ s

#7 Identify Subproblems 8.EE.A.1

There are

19

twos and

22

fives, so set aside

3

fives.

5^{22}=5^{19}\times5^3

19+3=22

.

2Pair twos with fives

#15 Organize Information in More Ways 8.EE.A.1

Powers with the same exponent multiply into one power of the product.

2^{19}\times5^{19}=(2\times5)^{19}=10^{19}

Each pair is a factor of

10

.

3Write as a number times a power of $10$

#15 Organize Information in More Ways 8.EE.A.3

The leftover is

5^3=125

.

2^{19}\times5^{22}=125\times10^{19}

5^3=125

.

4Count the digits

#9 Solve an Easier Related Problem 8.EE.A.3

Multiplying

125

by

10^{19}

writes

19

zeros after it:

125\times10^2=12500

has

3+2

digits, and in the same way this number has

3+19

digits.

n=3+19=22

Each factor of

10

adds exactly one digit.

Answer:

22

4 · Reviewdoes it hold up?

In scientific notation the number is $1.25\times10^{21}$ , which lies between $10^{21}$ (the smallest $22$ -digit number) and $10^{22}$ . So it has $22$ digits.

Another way: Pair all of it at once using scientific notation: $2^{19}\times5^{22}=1.25\times10^{21}$ , and a number $c\times10^{k}$ with $1\le c<10$ has $k+1$ digits, giving $22$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Splitting $5^{22}$ and combining $2^{19}\times5^{19}$ into $10^{19}$.
8.EE.A.3 Use numbers expressed in the form of a single digit times a power of 10 — Reading the size of a number written with a power of $10$.

💡Takeaway. Every

2

paired with a

5

is a

10

-- count the tens and you've counted the zeros.

Variant 8 medium

The number $2^{18}\times5^{23}$ is a natural number with $n$ digits. Find the value of $n$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Find how many digits the natural number $2^{18}\times5^{23}$ has.

Givens

The number is $2^{18}\times5^{23}$ .

Unknowns

The number of digits $n$ .

Constraints

Multiplying by $10$ adds one zero, so a number written as (a number) $\times10^k$ is easy to count.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Each $2\times5$ makes a $10$ . Pair up as many $2$ s with $5$ s as possible, leaving a small leftover number times a power of $10$ .

3 · Execute4 carry out the plan

1Split off the extra $5$ s

#7 Identify Subproblems 8.EE.A.1

There are

18

twos and

23

fives, so set aside

5

fives.

5^{23}=5^{18}\times5^5

18+5=23

.

2Pair twos with fives

#15 Organize Information in More Ways 8.EE.A.1

Powers with the same exponent multiply into one power of the product.

2^{18}\times5^{18}=(2\times5)^{18}=10^{18}

Each pair is a factor of

10

.

3Write as a number times a power of $10$

#15 Organize Information in More Ways 8.EE.A.3

The leftover is

5^5=3125

.

2^{18}\times5^{23}=3125\times10^{18}

5^5=3125

.

4Count the digits

#9 Solve an Easier Related Problem 8.EE.A.3

Multiplying

3125

by

10^{18}

writes

18

zeros after it:

3125\times10^2=312500

has

4+2

digits, and in the same way this number has

4+18

digits.

n=4+18=22

Each factor of

10

adds exactly one digit.

Answer:

22

4 · Reviewdoes it hold up?

In scientific notation the number is $3.125\times10^{21}$ , which lies between $10^{21}$ (the smallest $22$ -digit number) and $10^{22}$ . So it has $22$ digits.

Another way: Pair all of it at once using scientific notation: $2^{18}\times5^{23}=3.125\times10^{21}$ , and a number $c\times10^{k}$ with $1\le c<10$ has $k+1$ digits, giving $22$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Splitting $5^{23}$ and combining $2^{18}\times5^{18}$ into $10^{18}$.
8.EE.A.3 Use numbers expressed in the form of a single digit times a power of 10 — Reading the size of a number written with a power of $10$.

💡Takeaway. Every

2

paired with a

5

is a

10

-- count the tens and you've counted the zeros.

Variant 9 hard

The number $2^{24}\times5^{20}$ is a natural number with $n$ digits. Find the value of $n$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Find how many digits the natural number $2^{24}\times5^{20}$ has.

Givens

The number is $2^{24}\times5^{20}$ .

Unknowns

The number of digits $n$ .

Constraints

Multiplying by $10$ adds one zero, so a number written as (a number) $\times10^k$ is easy to count.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Each $2\times5$ makes a $10$ . Pair up as many $2$ s with $5$ s as possible, leaving a small leftover number times a power of $10$ .

3 · Execute4 carry out the plan

1Split off the extra $2$ s

#7 Identify Subproblems 8.EE.A.1

There are

24

twos and

20

fives, so set aside

4

twos.

2^{24}=2^{20}\times2^4

20+4=24

.

2Pair twos with fives

#15 Organize Information in More Ways 8.EE.A.1

Powers with the same exponent multiply into one power of the product.

2^{20}\times5^{20}=(2\times5)^{20}=10^{20}

Each pair is a factor of

10

.

3Write as a number times a power of $10$

#15 Organize Information in More Ways 8.EE.A.3

The leftover is

2^4=16

.

2^{24}\times5^{20}=16\times10^{20}

2^4=16

.

4Count the digits

#9 Solve an Easier Related Problem 8.EE.A.3

Multiplying

16

by

10^{20}

writes

20

zeros after it:

16\times10^2=1600

has

2+2

digits, and in the same way this number has

2+20

digits.

n=2+20=22

Each factor of

10

adds exactly one digit.

Answer:

22

4 · Reviewdoes it hold up?

In scientific notation the number is $1.6\times10^{21}$ , which lies between $10^{21}$ (the smallest $22$ -digit number) and $10^{22}$ . So it has $22$ digits.

Another way: Pair all of it at once using scientific notation: $2^{24}\times5^{20}=1.6\times10^{21}$ , and a number $c\times10^{k}$ with $1\le c<10$ has $k+1$ digits, giving $22$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Splitting $2^{24}$ and combining $2^{20}\times5^{20}$ into $10^{20}$.
8.EE.A.3 Use numbers expressed in the form of a single digit times a power of 10 — Reading the size of a number written with a power of $10$.

💡Takeaway. Every

2

paired with a

5

is a

10

-- count the tens and you've counted the zeros.

Variant 10 hard

The number $2^{25}\times5^{21}$ is a natural number with $n$ digits. Find the value of $n$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Find how many digits the natural number $2^{25}\times5^{21}$ has.

Givens

The number is $2^{25}\times5^{21}$ .

Unknowns

The number of digits $n$ .

Constraints

Multiplying by $10$ adds one zero, so a number written as (a number) $\times10^k$ is easy to count.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Each $2\times5$ makes a $10$ . Pair up as many $2$ s with $5$ s as possible, leaving a small leftover number times a power of $10$ .

3 · Execute4 carry out the plan

1Split off the extra $2$ s

#7 Identify Subproblems 8.EE.A.1

There are

25

twos and

21

fives, so set aside

4

twos.

2^{25}=2^{21}\times2^4

21+4=25

.

2Pair twos with fives

#15 Organize Information in More Ways 8.EE.A.1

Powers with the same exponent multiply into one power of the product.

2^{21}\times5^{21}=(2\times5)^{21}=10^{21}

Each pair is a factor of

10

.

3Write as a number times a power of $10$

#15 Organize Information in More Ways 8.EE.A.3

The leftover is

2^4=16

.

2^{25}\times5^{21}=16\times10^{21}

2^4=16

.

4Count the digits

#9 Solve an Easier Related Problem 8.EE.A.3

Multiplying

16

by

10^{21}

writes

21

zeros after it:

16\times10^2=1600

has

2+2

digits, and in the same way this number has

2+21

digits.

n=2+21=23

Each factor of

10

adds exactly one digit.

Answer:

23

4 · Reviewdoes it hold up?

In scientific notation the number is $1.6\times10^{22}$ , which lies between $10^{22}$ (the smallest $23$ -digit number) and $10^{23}$ . So it has $23$ digits.

Another way: Pair all of it at once using scientific notation: $2^{25}\times5^{21}=1.6\times10^{22}$ , and a number $c\times10^{k}$ with $1\le c<10$ has $k+1$ digits, giving $23$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Splitting $2^{25}$ and combining $2^{21}\times5^{21}$ into $10^{21}$.
8.EE.A.3 Use numbers expressed in the form of a single digit times a power of 10 — Reading the size of a number written with a power of $10$.

💡Takeaway. Every

2

paired with a

5

is a

10

-- count the tens and you've counted the zeros.

Variant 11 hard

The number $2^{20}\times5^{25}$ is a natural number with $n$ digits. Find the value of $n$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Find how many digits the natural number $2^{20}\times5^{25}$ has.

Givens

The number is $2^{20}\times5^{25}$ .

Unknowns

The number of digits $n$ .

Constraints

Multiplying by $10$ adds one zero, so a number written as (a number) $\times10^k$ is easy to count.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Each $2\times5$ makes a $10$ . Pair up as many $2$ s with $5$ s as possible, leaving a small leftover number times a power of $10$ .

3 · Execute4 carry out the plan

1Split off the extra $5$ s

#7 Identify Subproblems 8.EE.A.1

There are

20

twos and

25

fives, so set aside

5

fives.

5^{25}=5^{20}\times5^5

20+5=25

.

2Pair twos with fives

#15 Organize Information in More Ways 8.EE.A.1

Powers with the same exponent multiply into one power of the product.

2^{20}\times5^{20}=(2\times5)^{20}=10^{20}

Each pair is a factor of

10

.

3Write as a number times a power of $10$

#15 Organize Information in More Ways 8.EE.A.3

The leftover is

5^5=3125

.

2^{20}\times5^{25}=3125\times10^{20}

5^5=3125

.

4Count the digits

#9 Solve an Easier Related Problem 8.EE.A.3

Multiplying

3125

by

10^{20}

writes

20

zeros after it:

3125\times10^2=312500

has

4+2

digits, and in the same way this number has

4+20

digits.

n=4+20=24

Each factor of

10

adds exactly one digit.

Answer:

24

4 · Reviewdoes it hold up?

In scientific notation the number is $3.125\times10^{23}$ , which lies between $10^{23}$ (the smallest $24$ -digit number) and $10^{24}$ . So it has $24$ digits.

Another way: Pair all of it at once using scientific notation: $2^{20}\times5^{25}=3.125\times10^{23}$ , and a number $c\times10^{k}$ with $1\le c<10$ has $k+1$ digits, giving $24$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Splitting $5^{25}$ and combining $2^{20}\times5^{20}$ into $10^{20}$.
8.EE.A.3 Use numbers expressed in the form of a single digit times a power of 10 — Reading the size of a number written with a power of $10$.

💡Takeaway. Every

2

paired with a

5

is a

10

-- count the tens and you've counted the zeros.

Variant 12 hard

The number $2^{30}\times5^{23}$ is a natural number with $n$ digits. Find the value of $n$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Find how many digits the natural number $2^{30}\times5^{23}$ has.

Givens

The number is $2^{30}\times5^{23}$ .

Unknowns

The number of digits $n$ .

Constraints

Multiplying by $10$ adds one zero, so a number written as (a number) $\times10^k$ is easy to count.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Each $2\times5$ makes a $10$ . Pair up as many $2$ s with $5$ s as possible, leaving a small leftover number times a power of $10$ .

3 · Execute4 carry out the plan

1Split off the extra $2$ s

#7 Identify Subproblems 8.EE.A.1

There are

30

twos and

23

fives, so set aside

7

twos.

2^{30}=2^{23}\times2^7

23+7=30

.

2Pair twos with fives

#15 Organize Information in More Ways 8.EE.A.1

Powers with the same exponent multiply into one power of the product.

2^{23}\times5^{23}=(2\times5)^{23}=10^{23}

Each pair is a factor of

10

.

3Write as a number times a power of $10$

#15 Organize Information in More Ways 8.EE.A.3

The leftover is

2^7=128

.

2^{30}\times5^{23}=128\times10^{23}

2^7=128

.

4Count the digits

#9 Solve an Easier Related Problem 8.EE.A.3

Multiplying

128

by

10^{23}

writes

23

zeros after it:

128\times10^2=12800

has

3+2

digits, and in the same way this number has

3+23

digits.

n=3+23=26

Each factor of

10

adds exactly one digit.

Answer:

26

4 · Reviewdoes it hold up?

In scientific notation the number is $1.28\times10^{25}$ , which lies between $10^{25}$ (the smallest $26$ -digit number) and $10^{26}$ . So it has $26$ digits.

Another way: Pair all of it at once using scientific notation: $2^{30}\times5^{23}=1.28\times10^{25}$ , and a number $c\times10^{k}$ with $1\le c<10$ has $k+1$ digits, giving $26$ .

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Splitting $2^{30}$ and combining $2^{23}\times5^{23}$ into $10^{23}$.
8.EE.A.3 Use numbers expressed in the form of a single digit times a power of 10 — Reading the size of a number written with a power of $10$.

💡Takeaway. Every

2

paired with a

5

is a

10

-- count the tens and you've counted the zeros.

Pair 2s with 5s to make powers of ten · 12 practice problems

Generated variants — 12

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