To determine which expression is equivalent to [tex]\( 64^{\frac{1}{4}} \)[/tex], we will evaluate each given option and compare it to [tex]\( 64^{\frac{1}{4}} \)[/tex].
1. Evaluate [tex]\( 64^{\frac{1}{4}} \)[/tex]:
[tex]\[
64^{\frac{1}{4}} = \sqrt[4]{64}
\][/tex]
We know that [tex]\( 64 = 4^3 \)[/tex]. So:
[tex]\[
\sqrt[4]{64} = \sqrt[4]{4^3} = (4^3)^{\frac{1}{4}} = 4^{\frac{3}{4}}
\][/tex]
Evaluating [tex]\( 4^{\frac{3}{4}} \)[/tex]:
[tex]\[
4 = 2^2 \quad \text{so} \quad 4^{\frac{3}{4}} = (2^2)^{\frac{3}{4}} = 2^{2 \cdot \frac{3}{4}} = 2^{\frac{3}{2}} = 2 \cdot \sqrt{2} \approx 2.8284271247461903
\][/tex]
Hence, [tex]\( 64^{\frac{1}{4}} \approx 2.8284271247461903 \)[/tex].
2. Evaluate [tex]\( 2 \sqrt[4]{4} \)[/tex]:
[tex]\[
2 \sqrt[4]{4} = 2 \cdot 4^{\frac{1}{4}}
\][/tex]
We know [tex]\( 4 = 2^2 \)[/tex], so:
[tex]\[
4^{\frac{1}{4}} = (2^2)^{\frac{1}{4}} = 2^{\frac{1}{2}} = \sqrt{2} \approx 1.4142135623730951
\][/tex]
Therefore:
[tex]\[
2 \cdot 4^{\frac{1}{4}} = 2 \cdot \sqrt{2} \approx 2 \cdot 1.4142135623730951 = 2.8284271247461903
\][/tex]
Thus, [tex]\( 2 \sqrt[4]{4} \approx 2.8284271247461903 \)[/tex].
3. Evaluate [tex]\( 4 \)[/tex]:
[tex]\[
4 = 4
\][/tex]
This is a simple constant value.
4. Evaluate [tex]\( 16 \)[/tex]:
[tex]\[
16 = 16
\][/tex]
This is another constant value.
5. Evaluate [tex]\( 16 \sqrt[4]{4} \)[/tex]:
[tex]\[
16 \sqrt[4]{4} = 16 \cdot 4^{\frac{1}{4}}
\][/tex]
Again, using [tex]\( 4^{\frac{1}{4}} \approx 1.4142135623730951 \)[/tex]:
[tex]\[
16 \cdot 4^{\frac{1}{4}} = 16 \cdot 1.4142135623730951 \approx 22.627416997969522
\][/tex]
Thus, [tex]\( 16 \sqrt[4]{4} \approx 22.627416997969522 \)[/tex].
By comparing all the evaluated expressions, we observe that the value of [tex]\( 2 \sqrt[4]{4} \approx 2.8284271247461903 \)[/tex] is equivalent to [tex]\( 64^{\frac{1}{4}} \approx 2.8284271247461903 \)[/tex].
Therefore, the expression equivalent to [tex]\( 64^{\frac{1}{4}} \)[/tex] is:
[tex]\[
2 \sqrt[4]{4}
\][/tex]
