Question 5:

What is [tex]\( 2^{\frac{3}{2}} \)[/tex] equal to?

A. [tex]\( 2 \sqrt{8} \)[/tex]

B. [tex]\( \sqrt[3]{8} \)[/tex]

C. [tex]\( \sqrt[3]{16} \)[/tex]

D. [tex]\( 2 \sqrt{16} \)[/tex]

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Question 6:

(04.07 MC)

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Answer :

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]