Select the correct answer.

Which of the following is equal to this expression? [tex]$(256 \cdot 64)^{\frac{1}{4}}$[/tex]

A. [tex]4 \cdot \sqrt[4]{4}[/tex]

B. [tex]8 \cdot \sqrt[4]{2}[/tex]

C. [tex]8 \cdot \sqrt[4]{4}[/tex]

D. [tex]2 \cdot \sqrt[4]{2}[/tex]



Answer :

To find the equivalent expression for [tex]\((256 \cdot 64)^{\frac{1}{4}}\)[/tex], let us evaluate it step-by-step.

First, let's simplify the given expression:

1. Calculate the product inside the parentheses:
[tex]\[ 256 \cdot 64 = 16384 \][/tex]

2. Now, apply the fourth root to the product:
[tex]\[ (16384)^{\frac{1}{4}} \][/tex]

The fourth root of [tex]\(16384\)[/tex] is found to be [tex]\(11.313708498984761\)[/tex].

Next, let's evaluate each option to see which one yields the same result:

1. [tex]\( 4 \cdot \sqrt[4]{4} \)[/tex]
[tex]\[ \sqrt[4]{4} \approx 1.189207115002721 \][/tex]
[tex]\[ \text{So},\ 4 \cdot 1.189207115002721 \approx 4.756828460010884 \][/tex]

2. [tex]\( 8 \cdot \sqrt[4]{2} \)[/tex]
[tex]\[ \sqrt[4]{2} \approx 1.189207115002721 \][/tex]
[tex]\[ \text{So},\ 8 \cdot 1.189207115002721 \approx 9.513656920021768 \][/tex]

3. [tex]\( 8 \cdot \sqrt[4]{4} \)[/tex]
[tex]\[ \sqrt[4]{4} \approx 1.414213562373095 \][/tex]
[tex]\[ \text{So},\ 8 \cdot 1.414213562373095 \approx 11.313708498984761 \][/tex]

4. [tex]\( 2 \cdot \sqrt[4]{2} \)[/tex]
[tex]\[ \sqrt[4]{2} \approx 1.189207115002721 \][/tex]
[tex]\[ \text{So},\ 2 \cdot 1.189207115002721 \approx 2.378414230005442 \][/tex]

Comparing these results with the evaluated fourth root of [tex]\(16384\)[/tex], which is [tex]\(11.313708498984761\)[/tex], we see that:

[tex]\[ 8 \cdot \sqrt[4]{4} = 11.313708498984761 \][/tex]

Thus, the correct answer is:
[tex]\[ 8 \cdot \sqrt[4]{4} \][/tex]

Therefore, the expression [tex]\((256 \cdot 64)^{\frac{1}{4}}\)[/tex] is equal to:
[tex]\[ \boxed{8 \cdot \sqrt[4]{4}} \][/tex]