To determine what [tex]\(2^{\frac{4}{3}}\)[/tex] is equal to, let's first understand what this expression means and then compare it to the provided options.
1. Expression Analysis:
- Base: The base of the exponent is 2.
- Exponent: The exponent is [tex]\(\frac{4}{3}\)[/tex].
2. Calculation:
- To simplify [tex]\(2^{\frac{4}{3}}\)[/tex], it can be interpreted as taking the cube root of a number and then raising it to the 4th power, or vice versa.
- Therefore, [tex]\(2^{\frac{4}{3}}\)[/tex] can be written as [tex]\(\left(2^2\right)^{\frac{2}{3}}\)[/tex] or alternatively as [tex]\(\left(2^{\frac{1}{3}}\right)^4\)[/tex].
3. Result:
[tex]\(2^{\frac{4}{3}} \approx 2.5198420997897464\)[/tex]
4. Option Comparison:
Let's break down each of the options and see which one matches our result of approximately 2.5198420997897464:
- Option 1: [tex]\(\sqrt[3]{8}\)[/tex]
- Since [tex]\(8 = 2^3\)[/tex], [tex]\(\sqrt[3]{8} = 2\)[/tex].
- This does not match our result.
- Option 2: [tex]\(\sqrt[2]{8}\)[/tex]
- This is the square root of 8, written as [tex]\(\sqrt{8}\)[/tex].
- [tex]\(\sqrt{8} \approx 2.8284271247461903\)[/tex].
- This does not match our result.
- Option 3: [tex]\(\sqrt[3]{16}\)[/tex]
- Since [tex]\(16 = 2^4\)[/tex], [tex]\(\sqrt[3]{16} = (2^4)^{1/3} = 2^{4/3}\)[/tex].
- This matches our result, as [tex]\(\sqrt[3]{16} \approx 2.5198420997897464\)[/tex].
- Option 4: [tex]\(\sqrt[2]{16}\)[/tex]
- This is the square root of 16, written as [tex]\(\sqrt{16}\)[/tex].
- [tex]\(\sqrt{16} = 4\)[/tex].
- This does not match our result.
Therefore, among the provided options, [tex]\(2^{\frac{4}{3}}\)[/tex] is equal to [tex]\(\sqrt[3]{16}\)[/tex].
