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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

3 answers

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. \(4x^3\sqrt{2}\).

are you positive

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. \(4x^3\sqrt{2}\).