Solve 9/4sqrt{48}div(3/8sqrt{31/5})

$$\frac{ 9 }{ 4 } \sqrt{ 48 } \div ( \frac{ 3 }{ 8 } \sqrt{ 3 \frac { 1 } { 5 } } )$$

Evaluate

$$6\sqrt{15}\approx 23.237900077$$

Solution Steps

Factor 48=4^{2}\times 3. Rewrite the square root of the product \sqrt{4^{2}\times 3} as the product of square roots \sqrt{4^{2}}\sqrt{3}. Take the square root of 4^{2}.

$$\frac{\frac{9}{4}\times 4\sqrt{3}}{\frac{3}{8}\sqrt{\frac{3\times 5+1}{5}}}$$

Cancel out 4 and 4.

$$\frac{9\sqrt{3}}{\frac{3}{8}\sqrt{\frac{3\times 5+1}{5}}}$$

Multiply 3 and 5 to get 15.

$$\frac{9\sqrt{3}}{\frac{3}{8}\sqrt{\frac{15+1}{5}}}$$

Add 15 and 1 to get 16.

$$\frac{9\sqrt{3}}{\frac{3}{8}\sqrt{\frac{16}{5}}}$$

Rewrite the square root of the division \sqrt{\frac{16}{5}} as the division of square roots \frac{\sqrt{16}}{\sqrt{5}}.

$$\frac{9\sqrt{3}}{\frac{3}{8}\times \frac{\sqrt{16}}{\sqrt{5}}}$$

Calculate the square root of 16 and get 4.

$$\frac{9\sqrt{3}}{\frac{3}{8}\times \frac{4}{\sqrt{5}}}$$

Rationalize the denominator of \frac{4}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{9\sqrt{3}}{\frac{3}{8}\times \frac{4\sqrt{5}}{\left(\sqrt{5}\right)^{2}}}$$

The square of \sqrt{5} is 5.

$$\frac{9\sqrt{3}}{\frac{3}{8}\times \frac{4\sqrt{5}}{5}}$$

Multiply \frac{3}{8} times \frac{4\sqrt{5}}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{9\sqrt{3}}{\frac{3\times 4\sqrt{5}}{8\times 5}}$$

Cancel out 4 in both numerator and denominator.

$$\frac{9\sqrt{3}}{\frac{3\sqrt{5}}{2\times 5}}$$

Divide 9\sqrt{3} by \frac{3\sqrt{5}}{2\times 5} by multiplying 9\sqrt{3} by the reciprocal of \frac{3\sqrt{5}}{2\times 5}.

$$\frac{9\sqrt{3}\times 2\times 5}{3\sqrt{5}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{2\times 3\times 5\sqrt{3}}{\sqrt{5}}$$

Rationalize the denominator of \frac{2\times 3\times 5\sqrt{3}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{2\times 3\times 5\sqrt{3}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{2\times 3\times 5\sqrt{3}\sqrt{5}}{5}$$

Multiply 2 and 3 to get 6.

$$\frac{6\times 5\sqrt{3}\sqrt{5}}{5}$$

Multiply 6 and 5 to get 30.

$$\frac{30\sqrt{3}\sqrt{5}}{5}$$

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{30\sqrt{15}}{5}$$

Divide 30\sqrt{15} by 5 to get 6\sqrt{15}.

$$6\sqrt{15}$$