If x is a real number satisfying the exponential equation 4^{x+1} - 4^{x-1} = 120, what is… — GRE General Test

If $x$ is a real number satisfying the exponential equation $4^{x+1} - 4^{x-1} = 120$ , what is the value of $2^{2x + 1}$ ?

The value of

2^{2x + 1}

is 64.

Factoring

4^{x-1}

from

4^{x+1} - 4^{x-1}

gives

4^{x-1}(16 - 1) = 15 \cdot 4^{x-1} = 120

. Dividing by 15 yields

4^{x-1} = 8

. Rewriting with base 2 gives

(2^2)^{x-1} = 2^{2x-2} = 2^3

, so

2x - 2 = 3

, meaning

2x = 5

. Substituting

2x = 5

into

2^{2x+1}

gives

2^{5+1} = 2^6 = 64

Factor out

4^{x-1}

from the left side of the equation

4^{x+1} - 4^{x-1} = 120

4^{x-1}(4^2 - 1) = 120

, which simplifies to

4^{x-1}(15) = 120

Factoring out the lowest power of 4 allows simplification of the terms on the left side.

Divide both sides by 15 to isolate

4^{x-1}

4^{x-1} = \frac{120}{15} = 8

Isolating the exponential expression is necessary to solve for

x

Express both sides with a common base of 2.

(2^2)^{x-1} = 2^3 \implies 2^{2(x-1)} = 2^3 \implies 2^{2x - 2} = 2^3

Converting to a common prime base allows equating the exponents.

Equate exponents to solve for

2x

2x - 2 = 3 \implies 2x = 5

Since the bases are equal and positive, their exponents must be equal.

Substitute

2x = 5

into the target expression

2^{2x + 1}

2^{5 + 1} = 2^6 = 64

Evaluating the exact expression requested in the stem.

Factoring exponential expressions with variable exponents and converting bases

Estimated Time:1m 30s

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