If 2^{x-1} + 2^{x+1} = 320, what is the value of x? — State PSC Exam

If $2^{x-1} + 2^{x+1} = 320$ , what is the value of $x$ ?

7

Factoring

2^{x-1}

from the left side gives

2^{x-1}(1 + 2^2) = 5 \cdot 2^{x-1} = 320

. Dividing both sides by

5

yields

2^{x-1} = 64 = 2^6

. Equating the powers gives

x - 1 = 6

, which solves to

x = 7

Factor out the common exponential term from the left-hand side

2^{x-1}(1 + 2^2) = 320 \implies 2^{x-1}(1 + 4) = 320 \implies 5 \cdot 2^{x-1} = 320

Applying the distributive law of exponents to simplify the sum.

Divide both sides by 5

2^{x-1} = \frac{320}{5} = 64

Isolate the exponential term with base 2.

Express 64 as a power of 2 and equate the exponents

64 = 2^6 \implies x - 1 = 6 \implies x = 7

Since the bases are equal (

2

), the exponents must be equal.

Laws of Indices and Exponential Equations

Soru