Question

select the correct answer.
which of the following is equal to this expression?
$(256 \cdot 64)^\frac{1}{4}$
$8 \cdot \sqrt4{2}$
$2 \cdot \sqrt4{2}$
$4 \cdot \sqrt4{4}$
$8 \cdot \sqrt4{4}$

Explanation:

Step1: Rewrite numbers as powers of 2

$256=2^8$, $64=2^6$

Step2: Multiply the powers inside

$2^8 \cdot 2^6 = 2^{8+6}=2^{14}$

Step3: Apply the fourth root

$(2^{14})^\frac{1}{4}=2^\frac{14}{4}=2^\frac{7}{2}$

Step4: Split the exponent

$2^\frac{7}{2}=2^{3+\frac{1}{2}}=2^3 \cdot 2^\frac{1}{2}=8 \cdot 2^\frac{1}{2}$

Step5: Rewrite $2^\frac{1}{2}$ as fourth root

$2^\frac{1}{2}=2^\frac{2}{4}=\sqrt[4]{2^2}=\sqrt[4]{4}$

Answer:

D. $8 \cdot \sqrt[4]{4}$