Solve frac{sqrt{8}}{sqrt[8]{16}}

$$\frac{ \sqrt{ 8 } }{ \sqrt[ 8 ]{ 16 } }$$

Evaluate

$$2$$

Solution Steps

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{2\sqrt{2}}{\sqrt[8]{16}}$$

Rewrite \sqrt[8]{16} as \sqrt[8]{2^{4}}. Convert from radical to exponential form and cancel out 4 in the exponent. Convert back to radical form.

$$\sqrt[8]{16}=\sqrt[8]{2^{4}}=2^{\frac{4}{8}}=2^{\frac{1}{2}}=\sqrt{2}$$

Insert the obtained value back in the expression.

$$\frac{2\sqrt{2}}{\sqrt{2}}$$

Rationalize the denominator of \frac{2\sqrt{2}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{2\sqrt{2}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{2\sqrt{2}\sqrt{2}}{2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{2\times 2}{2}$$

Multiply 2 and 2 to get 4.

$$\frac{4}{2}$$

Divide 4 by 2 to get 2.

$$2$$

Factor

$$2$$