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]
