Question

solve the given equation. 2 = \left(8^{\frac{1}{2}}\
ight)\left(2^{\frac{x}{2}}\
ight) x = \square (type an integer or a simplified fraction.)

Explanation:

Step1: Rewrite 8 as power of 2

$8 = 2^3$, so $8^\frac{1}{2} = (2^3)^\frac{1}{2} = 2^\frac{3}{2}$

Step2: Multiply the exponential terms

Use $a^m \cdot a^n = a^{m+n}$:
$2^\frac{3}{2} \cdot 2^\frac{x}{2} = 2^{\frac{3+x}{2}}$

Step3: Set exponents equal (same base)

Since $2 = 2^1$, equate exponents:
$\frac{3+x}{2} = 1$

Step4: Solve for x

Multiply both sides by 2:
$3+x = 2$
Subtract 3 from both sides:
$x = 2 - 3$

Answer:

$-1$