If x is a real number such that 2^{x+1} + 2^{x-1} = 40, what is the value of the expressio… — GRE General Test

If $x$ is a real number such that $2^{x+1} + 2^{x-1} = 40$ , what is the value of the expression $(x+1)^{x-1}$ ?

125

Factoring out

2^x

from the given equation yields

2^x(2 + 0.5) = 40

, which gives

2.5 \cdot 2^x = 40

and

2^x = 16

. This determines that

x = 4

. Substituting

x = 4

into

(x+1)^{x-1}

gives

(4+1)^{3} = 5^3 = 125

Rewrite the given exponential terms with a common power of 2.

2^{x+1} = 2^x \cdot 2^1

and

2^{x-1} = 2^x \cdot 2^{-1} = \frac{2^x}{2}

Applying product rule for exponents allows factoring out

2^x

Factor out

2^x

and solve for

x

2^x \left(2 + \frac{1}{2}\right) = 40 \implies 2^x \cdot \frac{5}{2} = 40 \implies 2^x = 16 \implies x = 4

Combining fractional coefficients isolates the exponential term

2^x

Substitute

x = 4

into the target expression

(x+1)^{x-1}

(4+1)^{4-1} = 5^3 = 125

Simplifying the base and exponent yields the final numerical value.

Solving exponential equations using power distribution rules and factoring.

Soru