Question

find the cubic root or 96.
2^3√12
2^3√7
4√6
2√12

Answer 1

To find which of the options is equivalent to the cubic root of 96, we need to calculate \(\sqrt[3]{96}\) and then check which of the given options simplifies to that value.

First, we simplify \( \sqrt[3]{96} \):

1. Factor 96 into its prime factors:
\[
96 = 2^5 \times 3 = (2^3) \times (2^2 \times 3) = 8 \times 12
\]

2. Now, calculate the cubic root:
\[
\sqrt[3]{96} = \sqrt[3]{2^5 \cdot 3} = \sqrt[3]{2^5} \cdot \sqrt[3]{3} = 2^{5/3} \cdot 3^{1/3}
\]

3. We can rewrite \(2^{5/3}\) as:
\[
2^{5/3} = 2^{1 + 2/3} = 2^{1} \cdot 2^{2/3} = 2 \cdot \sqrt[3]{4}
\]

So, we have:
\[
\sqrt[3]{96} = 2 \cdot \sqrt[3]{4} \cdot \sqrt[3]{3}
\]

Now, we will evaluate each option:

1. **\( 2 \cdot \sqrt[3]{12} \):**
\[
2 \cdot \sqrt[3]{12} = 2 \cdot \sqrt[3]{(4 \cdot 3)} = 2 \cdot \sqrt[3]{4} \cdot \sqrt[3]{3}
\]
This is equal to \( \sqrt[3]{96} \).

2. **\( 2 \cdot \sqrt[3]{7} \):**
This does not equal \(\sqrt[3]{96}\).

3. **\( 4 \cdot \sqrt[3]{6} \):**
This equals:
\[
4 \cdot \sqrt[3]{(2 \cdot 3)} = 4 \cdot \sqrt[3]{6}
\]
This does not equal \(\sqrt[3]{96}\).

4. **\( 2 \cdot \sqrt{12} \):**
This equals:
\[
2 \cdot 2\sqrt{3} = 4\sqrt{3}
\]
This does not equal \(\sqrt[3]{96}\).

Therefore, the option that equals the cubic root of 96 is:

**\( 2 \cdot \sqrt[3]{12} \)**.