Let r be the number that results when both the base and the exponent of a^b are tripled, where a, b>0. If r equals the product of a^b and x^b where x>0, then x=

1993 AMC 12 Problem 10

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10.

Let

r

be the number that results when both the base and the exponent of

a^b

are tripled, where

a,

b\gt0.

r

equals the product of

a^b

and

x^b

where

x\gt0,

then

x=

3

3a^2

27a^2

2a^{3b}

3a^{2b}

Tripling both parts changes

a^b

(3a)^{3b}

Rewrite the equality so both sides are

b

th powers

We have

r=(3a)^{3b}=\bigl(27a^3\bigr)^b.

Also

r=a^bx^b=(ax)^b.

Positivity allows us to equate the bases, so

ax=27a^3

and

x=27a^2.

Thus the correct answer is C.