Question

simplify: $(4x^{2})^{\frac{1}{2}}$

Explanation:

Step1: Apply power - of - a - product rule

$(4x^{2})^{\frac{1}{2}}=4^{\frac{1}{2}}\times(x^{2})^{\frac{1}{2}}$

Step2: Calculate $4^{\frac{1}{2}}$

$4^{\frac{1}{2}}=\sqrt{4}=2$

Step3: Apply power - of - a - power rule

$(x^{2})^{\frac{1}{2}}=x^{2\times\frac{1}{2}} = x^{1}=x$

Step4: Multiply the results

$4^{\frac{1}{2}}\times(x^{2})^{\frac{1}{2}}=2x$

It seems there is a mistake in the provided options as the correct simplified form of $(4x^{2})^{\frac{1}{2}}$ is $2x$. If we assume the original problem was $(8x^{3})^{\frac{1}{3}}$:

Step1: Apply power - of - a - product rule

$(8x^{3})^{\frac{1}{3}}=8^{\frac{1}{3}}\times(x^{3})^{\frac{1}{3}}$

Step2: Calculate $8^{\frac{1}{3}}$

$8^{\frac{1}{3}}=\sqrt[3]{8}=2$

Step3: Apply power - of - a - power rule

$(x^{3})^{\frac{1}{3}}=x^{3\times\frac{1}{3}}=x$

Step4: Multiply the results

$8^{\frac{1}{3}}\times(x^{3})^{\frac{1}{3}} = 2x$

If the problem was $(8x^{6})^{\frac{1}{3}}$:

Step1: Apply power - of - a - product rule

$(8x^{6})^{\frac{1}{3}}=8^{\frac{1}{3}}\times(x^{6})^{\frac{1}{3}}$

Step2: Calculate $8^{\frac{1}{3}}$

$8^{\frac{1}{3}}=\sqrt[3]{8}=2$

Step3: Apply power - of - a - power rule

$(x^{6})^{\frac{1}{3}}=x^{6\times\frac{1}{3}}=x^{2}$

Step4: Multiply the results

$8^{\frac{1}{3}}\times(x^{6})^{\frac{1}{3}}=2x^{2}$

If the problem was $(8x^{9})^{\frac{1}{3}}$:

Step1: Apply power - of - a - product rule

$(8x^{9})^{\frac{1}{3}}=8^{\frac{1}{3}}\times(x^{9})^{\frac{1}{3}}$

Step2: Calculate $8^{\frac{1}{3}}$

$8^{\frac{1}{3}}=\sqrt[3]{8}=2$

Step3: Apply power - of - a - power rule

$(x^{9})^{\frac{1}{3}}=x^{9\times\frac{1}{3}}=x^{3}$

Step4: Multiply the results

$8^{\frac{1}{3}}\times(x^{9})^{\frac{1}{3}} = 2x^{3}$

Assuming it was $(8x^{9})^{\frac{1}{3}}$:

Answer:

$2x^{3}$