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

Same exponent, so match the bases

A number $S$ is what you get from $a^b$ $(a>0,\ b>0)$ when both its base and its exponent are multiplied by $3$ . The same number $S$ is also the product of $a^b$ and $x^b$ . Express $x$ in terms of $a$ . (Assume $x>0$ .)

Expressions

Your answer

How to solve

Strategy Convert to Algebra — Turn each sentence into an expression for S, then use the exponent laws to write both as a power with the same exponent b. Once the exponents match, the bases must match.

1STEP 1

Write the first description

Tripling the base of a^b gives 3a; tripling the exponent gives 3b.

S=(3a)^(3b)

2STEP 2

Make the exponent b appear

Split the exponent 3b as 3 × b and use the power-of-a-power rule, then cube 3a.

(3a)^(3b)={(3a)³}^b=(27a³)^b

3STEP 3

Write the second description

Two powers with the same exponent multiply into one power of the product.

S=a^b × x^b=(ax)^b

4STEP 4

Compare the bases

S is now a b-th power two ways. With positive bases and b ＞ 0, equal powers need equal bases.

(27a³)^b=(ax)^b → 27a³=ax

5STEP 5

Solve for x

Divide both sides by a, which is not 0.

x=27a³/a=27a²

Answer

x=27a²

Try a=2, b=1: the first description gives S=6³=216, and x=27×2²=108 gives a^b × x^b=2×108=216. They agree.

Takeaway

When two powers share an exponent, line them up and just compare the bases.

Write the first description
Make the exponent b appear
Write the second description
Compare the bases
Solve for x

Where next?

Level upsuggested

Pair 2s with 5s to make powers of ten Rewrite in base 2 and factor the sum

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12 practice problems

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Make every prime exponent even Count numerators that cancel the denominator's 3s

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▶ Practice — 12 problems