If 5^{x+1} + 5^{x-1} = 650, what is the value of 2^{2x - 1}? — State PSC Exam

If $5^{x+1} + 5^{x-1} = 650$ , what is the value of $2^{2x - 1}$ ?

Factoring out

5^{x-1}

yields

26 \cdot 5^{x-1} = 650

, which simplifies to

5^{x-1} = 25

, giving

x = 3

. Substituting

x = 3

into

2^{2x - 1}

gives

2^5 = 32

Factor out common exponential terms from the given equation.

5^{x-1}(5^2 + 1) = 650 \implies 5^{x-1}(25 + 1) = 650 \implies 26 \cdot 5^{x-1} = 650

Use the product law of indices

5^{x+1} = 5^{x-1} \cdot 5^2

to rewrite the expression.

Solve for the unknown variable

x

5^{x-1} = \frac{650}{26} = 25 = 5^2 \implies x - 1 = 2 \implies x = 3

Since the bases are identical, equate the exponents.

Substitute

x = 3

into the expression

2^{2x - 1}

2^{2(3) - 1} = 2^{6 - 1} = 2^5 = 32

Evaluate the power after computing the exponent value.

Solving Exponential Equations using Laws of Indices

Question