Which is equivalent to 6 Superscript 3-halves Baseline?

Question

Answer options with 5 options
A.
StartRoot 18 EndRoot

B.
3 StartRoot 6 EndRoot

C.
6 StartRoot 6 EndRoot

D.
RootIndex 3 StartRoot 12 EndRoot

E.
RootIndex 3 StartRoot 36 EndRoot

Answer 1

To evaluate \( 6^{\frac{3}{2}} \), we can break it down into two parts.

1. First, we handle the exponentiation: \[ 6^{\frac{3}{2}} = (6^3)^{\frac{1}{2}} \]

2. Now, calculate \( 6^3 \): \[ 6^3 = 216 \]

3. Now, take the square root: \[ (6^3)^{\frac{1}{2}} = \sqrt{216} \]

Next, we can simplify \( \sqrt{216} \): \[ 216 = 36 \times 6 \] Thus, \[ \sqrt{216} = \sqrt{36 \times 6} = \sqrt{36} \cdot \sqrt{6} = 6\sqrt{6} \]

Now we can compare this result with the answer options:

A. \( \sqrt{18} \)
B. \( 3\sqrt{6} \)
C. \( 6\sqrt{6} \)
D. \( \sqrt[3]{12} \)
E. \( \sqrt[3]{36} \)

The correct equivalent of \( 6^{\frac{3}{2}} \) is: C. \( 6\sqrt{6} \).