Solve sqrt{2^3*3}divsqrt{2*3}*1/sqrt{2*3}

$$\sqrt{ { 2 }^{ 3 } \times 3 } \div \sqrt{ 2 \times 3 } \times \frac{ 1 }{ \sqrt{ 2 \times 3 } }$$

Evaluate

$$\frac{\sqrt{6}}{3}\approx 0.816496581$$

Solution Steps

Calculate 2 to the power of 3 and get 8.

$$\frac{\sqrt{8\times 3}}{\sqrt{2\times 3}}\times \frac{1}{\sqrt{2\times 3}}$$

Multiply 8 and 3 to get 24.

$$\frac{\sqrt{24}}{\sqrt{2\times 3}}\times \frac{1}{\sqrt{2\times 3}}$$

Factor 24=2^{2}\times 6. Rewrite the square root of the product \sqrt{2^{2}\times 6} as the product of square roots \sqrt{2^{2}}\sqrt{6}. Take the square root of 2^{2}.

$$\frac{2\sqrt{6}}{\sqrt{2\times 3}}\times \frac{1}{\sqrt{2\times 3}}$$

Multiply 2 and 3 to get 6.

$$\frac{2\sqrt{6}}{\sqrt{6}}\times \frac{1}{\sqrt{2\times 3}}$$

Rationalize the denominator of \frac{2\sqrt{6}}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{2\sqrt{6}\sqrt{6}}{\left(\sqrt{6}\right)^{2}}\times \frac{1}{\sqrt{2\times 3}}$$

The square of \sqrt{6} is 6.

$$\frac{2\sqrt{6}\sqrt{6}}{6}\times \frac{1}{\sqrt{2\times 3}}$$

Multiply \sqrt{6} and \sqrt{6} to get 6.

$$\frac{2\times 6}{6}\times \frac{1}{\sqrt{2\times 3}}$$

Multiply 2 and 6 to get 12.

$$\frac{12}{6}\times \frac{1}{\sqrt{2\times 3}}$$

Divide 12 by 6 to get 2.

$$2\times \frac{1}{\sqrt{2\times 3}}$$

Multiply 2 and 3 to get 6.

$$2\times \frac{1}{\sqrt{6}}$$

Rationalize the denominator of \frac{1}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$2\times \frac{\sqrt{6}}{\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$2\times \frac{\sqrt{6}}{6}$$

Cancel out 6, the greatest common factor in 2 and 6.

$$\frac{\sqrt{6}}{3}$$