Question

Which expression is equivalent to square root of 32x^6 ?

A. 4x^3 square root of 2

B. 8x^3

C. 4x square root of 2

D. 16x^3

Answer 1

To simplify the expression \(\sqrt{32x^6}\), we can break it down as follows:

1. **Simplify the constant**:
\[
32 = 16 \times 2 = 4^2 \times 2
\]
Thus,
\[
\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}
\]

2. **Simplify the variable**:
When simplifying \(\sqrt{x^6}\), we can use the property that \(\sqrt{x^n} = x^{n/2}\):
\[
\sqrt{x^6} = x^{6/2} = x^3
\]

3. **Combine the results**:
Therefore, we have:
\[
\sqrt{32x^6} = \sqrt{32} \times \sqrt{x^6} = (4\sqrt{2}) \times (x^3) = 4x^3\sqrt{2}
\]

So, the equivalent expression is **A. \(4x^3\sqrt{2}\)**.

Answer 2

are you positive

Answer 3

Yes, I'm confident in my previous calculations. Let's recap the simplification of \(\sqrt{32x^6}\) to verify:

1. **Calculate the square root of the constant**:
- \(32\) can be expressed as \(16 \times 2\).
- Thus, \(\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}\).

2. **Calculate the square root of the variable**:
- For \(x^6\), we apply the property of square roots: \(\sqrt{x^n} = x^{n/2}\).
- So, \(\sqrt{x^6} = x^{6/2} = x^3\).

3. **Combine both results**:
- Now we combine the results:
\[
\sqrt{32x^6} = \sqrt{32} \times \sqrt{x^6} = (4\sqrt{2}) \times (x^3) = 4x^3\sqrt{2}.
\]

Thus, the expression \(\sqrt{32x^6}\) simplifies to \(4x^3\sqrt{2}\), confirming that the answer is indeed **A. \(4x^3\sqrt{2}\)**.