If 9^{x+1} - 9^x = 216, what is the value of 4^x? — GMAT

If $9^{x+1} - 9^x = 216$ , what is the value of $4^x$ ?

Answer: 8

The value of

4^x

is 8.

Factoring

9^x

from

9^{x+1} - 9^x

gives

9^x(9 - 1) = 216

, or

8 \cdot 9^x = 216

. Dividing by 8 yields

9^x = 27

. Expressing both sides with base 3 gives

3^{2x} = 3^3

, so

2x = 3

and

x = \frac{3}{2}

. Substituting this value into

4^x

results in

4^{3/2} = (\sqrt{4})^3 = 2^3 = 8

Factor out common exponent terms from the left side of the equation

9^x(9^1 - 1) = 216 \implies 8 \cdot 9^x = 216

Using the exponent property

a^{m+n} = a^m \cdot a^n

, rewrite

9^{x+1}

9^x \cdot 9^1

to factor out

9^x

Isolate the exponential term

9^x = 27

Dividing both sides of

8 \cdot 9^x = 216

by 8 yields

27

Convert both sides to a common prime base of 3

(3^2)^x = 3^3 \implies 3^{2x} = 3^3 \implies 2x = 3 \implies x = \frac{3}{2}

Since bases are equal, exponents must be equal.

Evaluate the target expression

4^x

4^{3/2} = (4^{1/2})^3 = 2^3 = 8

Substitute

x = \frac{3}{2}

into

4^x

and apply the rule

a^{m/n} = (\sqrt[n]{a})^m

Solving exponential equations by factoring and equating powers with a common base.

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