To determine the value of \( 2^{\frac{4}{3}} \), we must understand the steps involved in evaluating this expression. To further analyze, we will also evaluate and compare the provided options:
1. Evaluate \(2^{\frac{4}{3}}\):
- To find \(2^{\frac{4}{3}}\), we can rewrite it as \((2^4)^{\frac{1}{3}}\).
- Calculating \(2^4\):
[tex]\[
2^4 = 2 \times 2 \times 2 \times 2 = 16
\][/tex]
- Now, take the cube root of 16:
[tex]\[
(16)^{\frac{1}{3}} \approx 2.5198420997897464
\][/tex]
So, \(2^{\frac{4}{3}} \approx 2.5198420997897464\).
2. Evaluate the options:
- Option 1: \(\sqrt[3]{8}\)
[tex]\[
\sqrt[3]{8} = 8^{\frac{1}{3}} = 2.0
\][/tex]
- Option 2: \(2 \sqrt{8}\)
- Calculate \(\sqrt{8}\):
[tex]\[
\sqrt{8} = (8)^{\frac{1}{2}} = \sqrt{(4 \cdot 2)} = 2\sqrt{2} \approx 2 \times 2.8284271247461903 \approx 5.656854249492381
\][/tex]
- Option 3: \(\sqrt[3]{16}\)
- Calculate \((16)^{\frac{1}{3}}\):
[tex]\[
\sqrt[3]{16} \approx 2.5198420997897464
\][/tex]
- Option 4: \(\sqrt{16}\)
- Calculate \((16)^{\frac{1}{2}}\):
[tex]\[
\sqrt{16} = 4.0
\][/tex]
From the above evaluations, we see that:
- \( 2^{\frac{4}{3}} \approx 2.5198420997897464 \)
- The value of \(\sqrt[3]{16} \approx 2.5198420997897464\)
Therefore, [tex]\( 2^{\frac{4}{3}} \)[/tex] is equal to [tex]\(\sqrt[3]{16}\)[/tex].
