To find which of the given expressions is equivalent to [tex]\(16^{\frac{3}{2}}\)[/tex], we will evaluate each expression step by step:
1. Evaluate [tex]\(16^{\frac{3}{2}}\)[/tex]:
- The expression [tex]\(16^{\frac{3}{2}}\)[/tex] represents a power of 16. The exponent [tex]\(\frac{3}{2}\)[/tex] can be interpreted as follows:
[tex]\[
16^{\frac{3}{2}} = (16^{\frac{1}{2}})^3 = (\sqrt{16})^3.
\][/tex]
- Since [tex]\( \sqrt{16} = 4 \)[/tex], we have:
[tex]\[
(16^{\frac{1}{2}})^3 = 4^3 = 64.
\][/tex]
- Therefore, [tex]\(16^{\frac{3}{2}} = 64.0\)[/tex].
2. Evaluate [tex]\(\sqrt{16^3}\)[/tex]:
- The expression [tex]\(\sqrt{16^3}\)[/tex] represents the square root of 16 raised to the power of 3. We can calculate it as follows:
[tex]\[
\sqrt{16^3} = \sqrt{(16 \cdot 16 \cdot 16)}.
\][/tex]
- First, find [tex]\(16^3\)[/tex]:
[tex]\[
16^3 = 16 \cdot 16 \cdot 16 = 4096.
\][/tex]
- Then, find the square root of 4096:
[tex]\[
\sqrt{4096} = 64.
\][/tex]
- Therefore, [tex]\(\sqrt{16^3} = 64.0\)[/tex].
3. Evaluate [tex]\(16 - \frac{3}{2}\)[/tex]:
- The expression [tex]\(16 - \frac{3}{2}\)[/tex] can be calculated as follows:
[tex]\[
16 - \frac{3}{2} = 16 - 1.5 = 14.5.
\][/tex]
- Therefore, [tex]\(16 - \frac{3}{2} = 14.5\)[/tex].
4. Evaluate [tex]\(16 - \frac{2}{3}\)[/tex]:
- The expression [tex]\(16 - \frac{2}{3}\)[/tex] can be calculated as follows:
[tex]\[
16 - \frac{2}{3} = 16 - 0.666666\ldots \approx 15.333333\ldots.
\][/tex]
- Therefore, [tex]\(16 - \frac{2}{3} = 15.333333\ldots\)[/tex].
5. Evaluate [tex]\(\sqrt[3]{16^2}\)[/tex]:
- The expression [tex]\(\sqrt[3]{16^2}\)[/tex] represents the cube root of 16 squared. We can calculate it as follows:
[tex]\[
\sqrt[3]{16^2} = \sqrt[3]{(16 \cdot 16)}.
\][/tex]
- First, find [tex]\(16^2\)[/tex]:
[tex]\[
16^2 = 16 \cdot 16 = 256.
\][/tex]
- Then, find the cube root of 256:
[tex]\[
\sqrt[3]{256} \approx 6.35.
\][/tex]
- Therefore, [tex]\(\sqrt[3]{16^2} \approx 6.35\)[/tex].
Comparing these evaluations, we see that [tex]\(16^{\frac{3}{2}} = 64\)[/tex] is equivalent to [tex]\(\sqrt{16^3} = 64\)[/tex]. Therefore, the correct answer is:
[tex]\[
\sqrt{16^3}.
\][/tex]
