If \frac{8^x \cdot 4^{x-1}}{16^{x+1}} = 32, what is the value of x? — GMAT

If $\frac{8^x \cdot 4^{x-1}}{16^{x+1}} = 32$ , what is the value of $x$ ?

Answer: 11

The value of

x

is 11.

Rewriting all bases as powers of 2 yields

\frac{2^{3x} \cdot 2^{2x-2}}{2^{4x+4}} = 2^5

. Simplifying the left side gives

2^{x-6} = 2^5

. Equating exponents results in

x - 6 = 5

, giving

x = 11

Express all bases as powers of 2.

8 = 2^3

4 = 2^2

16 = 2^4

, and

32 = 2^5

Converting to a common base allows exponents to be combined using standard exponent laws.

Simplify the numerator and denominator using the power rule

(a^m)^n = a^{mn}

and product rule

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

Numerator:

2^{3x} \cdot 2^{2x-2} = 2^{5x-2}

. Denominator:

2^{4(x+1)} = 2^{4x+4}

Distribute exponents carefully when multiplying powers with equal bases.

Apply the quotient rule

\frac{a^m}{a^n} = a^{m-n}

to combine the left side into a single power of 2.

\frac{2^{5x-2}}{2^{4x+4}} = 2^{(5x-2) - (4x+4)} = 2^{x-6}

Subtract the denominator's exponent from the numerator's exponent.

Equate exponents and solve for

x

2^{x-6} = 2^5 \implies x - 6 = 5 \implies x = 11

Since the bases are identical (

2

), the exponent expressions must be equal.

Solving exponential equations by expressing all numbers with a common base and applying exponent rules.

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