Solve 3sqrt{1/512}divsqrt{1/64}3sqrt{1/512}divsqrt{1/64}

$$3 \sqrt{ \frac{ 1 }{ 512 } } \div \sqrt{ \frac{ 1 }{ 64 } } 3 \sqrt{ \frac{ 1 }{ 512 } } \div \sqrt{ \frac{ 1 }{ 64 } }$$

Evaluate

$$\frac{9}{8}=1.125$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{1}{512}} as the division of square roots \frac{\sqrt{1}}{\sqrt{512}}.

$$\frac{3\times \frac{3\times \frac{\sqrt{1}}{\sqrt{512}}}{\sqrt{\frac{1}{64}}}\sqrt{\frac{1}{512}}}{\sqrt{\frac{1}{64}}}$$

Calculate the square root of 1 and get 1.

$$\frac{3\times \frac{3\times \frac{1}{\sqrt{512}}}{\sqrt{\frac{1}{64}}}\sqrt{\frac{1}{512}}}{\sqrt{\frac{1}{64}}}$$

Factor 512=16^{2}\times 2. Rewrite the square root of the product \sqrt{16^{2}\times 2} as the product of square roots \sqrt{16^{2}}\sqrt{2}. Take the square root of 16^{2}.

$$\frac{3\times \frac{3\times \frac{1}{16\sqrt{2}}}{\sqrt{\frac{1}{64}}}\sqrt{\frac{1}{512}}}{\sqrt{\frac{1}{64}}}$$

Rationalize the denominator of \frac{1}{16\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{3\times \frac{3\times \frac{\sqrt{2}}{16\left(\sqrt{2}\right)^{2}}}{\sqrt{\frac{1}{64}}}\sqrt{\frac{1}{512}}}{\sqrt{\frac{1}{64}}}$$

The square of \sqrt{2} is 2.

$$\frac{3\times \frac{3\times \frac{\sqrt{2}}{16\times 2}}{\sqrt{\frac{1}{64}}}\sqrt{\frac{1}{512}}}{\sqrt{\frac{1}{64}}}$$

Multiply 16 and 2 to get 32.

$$\frac{3\times \frac{3\times \frac{\sqrt{2}}{32}}{\sqrt{\frac{1}{64}}}\sqrt{\frac{1}{512}}}{\sqrt{\frac{1}{64}}}$$

Express 3\times \frac{\sqrt{2}}{32} as a single fraction.

$$\frac{3\times \frac{\frac{3\sqrt{2}}{32}}{\sqrt{\frac{1}{64}}}\sqrt{\frac{1}{512}}}{\sqrt{\frac{1}{64}}}$$

Rewrite the square root of the division \frac{1}{64} as the division of square roots \frac{\sqrt{1}}{\sqrt{64}}. Take the square root of both numerator and denominator.

$$\frac{3\times \frac{\frac{3\sqrt{2}}{32}}{\frac{1}{8}}\sqrt{\frac{1}{512}}}{\sqrt{\frac{1}{64}}}$$

Divide \frac{3\sqrt{2}}{32} by \frac{1}{8} by multiplying \frac{3\sqrt{2}}{32} by the reciprocal of \frac{1}{8}.

$$\frac{3\times \frac{3\sqrt{2}\times 8}{32}\sqrt{\frac{1}{512}}}{\sqrt{\frac{1}{64}}}$$

Multiply 3 and 8 to get 24.

$$\frac{3\times \frac{24\sqrt{2}}{32}\sqrt{\frac{1}{512}}}{\sqrt{\frac{1}{64}}}$$

Divide 24\sqrt{2} by 32 to get \frac{3}{4}\sqrt{2}.

$$\frac{3\times \frac{3}{4}\sqrt{2}\sqrt{\frac{1}{512}}}{\sqrt{\frac{1}{64}}}$$

Express 3\times \frac{3}{4} as a single fraction.

$$\frac{\frac{3\times 3}{4}\sqrt{2}\sqrt{\frac{1}{512}}}{\sqrt{\frac{1}{64}}}$$

Multiply 3 and 3 to get 9.

$$\frac{\frac{9}{4}\sqrt{2}\sqrt{\frac{1}{512}}}{\sqrt{\frac{1}{64}}}$$

Rewrite the square root of the division \sqrt{\frac{1}{512}} as the division of square roots \frac{\sqrt{1}}{\sqrt{512}}.

$$\frac{\frac{9}{4}\sqrt{2}\times \frac{\sqrt{1}}{\sqrt{512}}}{\sqrt{\frac{1}{64}}}$$

Calculate the square root of 1 and get 1.

$$\frac{\frac{9}{4}\sqrt{2}\times \frac{1}{\sqrt{512}}}{\sqrt{\frac{1}{64}}}$$

Factor 512=16^{2}\times 2. Rewrite the square root of the product \sqrt{16^{2}\times 2} as the product of square roots \sqrt{16^{2}}\sqrt{2}. Take the square root of 16^{2}.

$$\frac{\frac{9}{4}\sqrt{2}\times \frac{1}{16\sqrt{2}}}{\sqrt{\frac{1}{64}}}$$

Rationalize the denominator of \frac{1}{16\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\frac{9}{4}\sqrt{2}\times \frac{\sqrt{2}}{16\left(\sqrt{2}\right)^{2}}}{\sqrt{\frac{1}{64}}}$$

The square of \sqrt{2} is 2.

$$\frac{\frac{9}{4}\sqrt{2}\times \frac{\sqrt{2}}{16\times 2}}{\sqrt{\frac{1}{64}}}$$

Multiply 16 and 2 to get 32.

$$\frac{\frac{9}{4}\sqrt{2}\times \frac{\sqrt{2}}{32}}{\sqrt{\frac{1}{64}}}$$

Multiply \frac{9}{4} times \frac{\sqrt{2}}{32} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{9\sqrt{2}}{4\times 32}\sqrt{2}}{\sqrt{\frac{1}{64}}}$$

Rewrite the square root of the division \frac{1}{64} as the division of square roots \frac{\sqrt{1}}{\sqrt{64}}. Take the square root of both numerator and denominator.

$$\frac{\frac{9\sqrt{2}}{4\times 32}\sqrt{2}}{\frac{1}{8}}$$

Multiply 4 and 32 to get 128.

$$\frac{\frac{9\sqrt{2}}{128}\sqrt{2}}{\frac{1}{8}}$$

Express \frac{9\sqrt{2}}{128}\sqrt{2} as a single fraction.

$$\frac{\frac{9\sqrt{2}\sqrt{2}}{128}}{\frac{1}{8}}$$

Divide \frac{9\sqrt{2}\sqrt{2}}{128} by \frac{1}{8} by multiplying \frac{9\sqrt{2}\sqrt{2}}{128} by the reciprocal of \frac{1}{8}.

$$\frac{9\sqrt{2}\sqrt{2}\times 8}{128}$$

Divide 9\sqrt{2}\sqrt{2}\times 8 by 128 to get 9\sqrt{2}\sqrt{2}\times \frac{1}{16}.

$$9\sqrt{2}\sqrt{2}\times \frac{1}{16}$$

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

$$9\times 2\times \frac{1}{16}$$

Multiply 9 and 2 to get 18.

$$18\times \frac{1}{16}$$

Multiply 18 and \frac{1}{16} to get \frac{18}{16}.

$$\frac{18}{16}$$

Reduce the fraction \frac{18}{16} to lowest terms by extracting and canceling out 2.

$$\frac{9}{8}$$

Factor

$$\frac{3 ^ {2}}{2 ^ {3}} = 1\frac{1}{8} = 1.125$$