All right, let's solve the expression [tex]\(2^{\frac{3}{2}}\)[/tex].
The exponent [tex]\(\frac{3}{2}\)[/tex] can be interpreted as a combination of an exponent and a root. Specifically, [tex]\(a^{\frac{m}{n}}\)[/tex] means the [tex]\(n\)[/tex]-th root of [tex]\(a^m\)[/tex].
1. Step 1: Interpret the exponent [tex]\(\frac{3}{2}\)[/tex]:
[tex]\[
2^{\frac{3}{2}} = \left(2^3\right)^{\frac{1}{2}}
\][/tex]
This can also be written as:
[tex]\[
2^{\frac{3}{2}} = \sqrt{2^3}
\][/tex]
2. Step 2: Compute [tex]\(2^3\)[/tex]:
[tex]\[
2^3 = 8
\][/tex]
3. Step 3: Take the square root of 8:
[tex]\[
\sqrt{8}
\][/tex]
Therefore, [tex]\(2^{\frac{3}{2}} = \sqrt{8}\)[/tex].
The correct option from the given choices is:
- [tex]\( \sqrt[3]{8} \)[/tex]
- [tex]\( 2 \sqrt{8} \)[/tex]
- [tex]\( \sqrt[3]{16} \)[/tex]
- [tex]\( 2 \sqrt{16} \)[/tex]
As we determined, [tex]\(2^{\frac{3}{2}} = \sqrt{8}\)[/tex].
Let's match this with the options:
- [tex]\( \frac{3}{2}\)[/tex]-th power of 2 is not [tex]\( \sqrt[3]{8}\)[/tex] (cube root of 8).
- [tex]\(2 \sqrt{8}\)[/tex] directly matches the result we got: [tex]\(\sqrt{8}\)[/tex].
- [tex]\( \frac{3}{2}\)[/tex]-th power of 2 is not [tex]\( \sqrt[3]{16}\)[/tex].
- [tex]\( \frac{3}{2}\)[/tex]-th power of 2 is not [tex]\(2 \sqrt{16}\)[/tex].
Hence, the correct answer is:
[tex]\[ 2 \sqrt{8} \][/tex]
