Numbers & Place Value

Depth · one concept from first principles · step 2 of 4

Multiply top and bottom to use the given power

Use the rule a−n=1ana^{-n}=\frac{1}{a^n} for every natural number nn. If a−2=3a^{-2}=3, find the value of a3−a−3a3+a−3\frac{a^3-a^{-3}}{a^3+a^{-3}}.

ExpressionsNumber system
Your answer
How to solve
Strategy Convert to Algebra — We cannot easily find a itself, but we know a⁻². Multiply the top and bottom by a⁻³ so that every term becomes either 1 or a⁻⁶, which is a power of a⁻².
1STEP 1

Turn the given into a⁻⁶

Cube the given value: a⁻⁶=1/a⁶=(1/a²)³.

a⁻⁶=(a⁻²)³=3³=27
2STEP 2

Multiply top and bottom by a⁻³

This keeps the value of the fraction. Each a³ × a⁻³ becomes a⁰=1, and each a⁻³ × a⁻³ becomes a⁻⁶.

(a³-a⁻³)/(a³+a⁻³)=(1-a⁻⁶)/(1+a⁻⁶)
3STEP 3

Substitute and simplify

Put a⁻⁶=27 into the new fraction.

(1-27)/(1+27)=-26/28=-13/14
Answer
-13/14
Since a⁻²=3, a²=1/3 and a⁶=1/27. Multiplying the numerator and denominator by a³ gives (a⁶-1)/(a⁶+1)=(1/27-1)/(1/27+1)=-13/14. This checks the result whether a is positive or negative.
Takeaway

If you know one power of a, reshape the expression so only that power appears.

  • Turn the given into a⁻⁶
  • Multiply top and bottom by a⁻³
  • Substitute and simplify
Where next?
Another one like thissuggested
12 practice problems
Step back and solidifysuggested
Rewrite in base 2 and factor the sumPair 2s with 5s to make powers of ten

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▶ Practice — 12 problems