Solve 9/4sqrt{48}div(3/8sqrt{21/5})

$$\frac { 9 } { 4 } \sqrt { 48 } \div ( \frac { 3 } { 8 } \sqrt { 2 \frac { 1 } { 5 } } )$$

Evaluate

$$\frac{24\sqrt{165}}{11}\approx 28.02596199$$

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}.

$$\frac{\frac{9}{4}\times 4\sqrt{3}}{\frac{3}{8}\sqrt{\frac{2\times 5+1}{5}}}$$

Cancel out 4 and 4.

$$\frac{9\sqrt{3}}{\frac{3}{8}\sqrt{\frac{2\times 5+1}{5}}}$$

Multiply 2 and 5 to get 10.

$$\frac{9\sqrt{3}}{\frac{3}{8}\sqrt{\frac{10+1}{5}}}$$

Add 10 and 1 to get 11.

$$\frac{9\sqrt{3}}{\frac{3}{8}\sqrt{\frac{11}{5}}}$$

Rewrite the square root of the division \sqrt{\frac{11}{5}} as the division of square roots \frac{\sqrt{11}}{\sqrt{5}}.

$$\frac{9\sqrt{3}}{\frac{3}{8}\times \frac{\sqrt{11}}{\sqrt{5}}}$$

Rationalize the denominator of \frac{\sqrt{11}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{9\sqrt{3}}{\frac{3}{8}\times \frac{\sqrt{11}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}}$$

The square of \sqrt{5} is 5.

$$\frac{9\sqrt{3}}{\frac{3}{8}\times \frac{\sqrt{11}\sqrt{5}}{5}}$$

To multiply \sqrt{11} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{9\sqrt{3}}{\frac{3}{8}\times \frac{\sqrt{55}}{5}}$$

Multiply \frac{3}{8} times \frac{\sqrt{55}}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{9\sqrt{3}}{\frac{3\sqrt{55}}{8\times 5}}$$

Divide 9\sqrt{3} by \frac{3\sqrt{55}}{8\times 5} by multiplying 9\sqrt{3} by the reciprocal of \frac{3\sqrt{55}}{8\times 5}.

$$\frac{9\sqrt{3}\times 8\times 5}{3\sqrt{55}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{3\times 5\times 8\sqrt{3}}{\sqrt{55}}$$

Rationalize the denominator of \frac{3\times 5\times 8\sqrt{3}}{\sqrt{55}} by multiplying numerator and denominator by \sqrt{55}.

$$\frac{3\times 5\times 8\sqrt{3}\sqrt{55}}{\left(\sqrt{55}\right)^{2}}$$

The square of \sqrt{55} is 55.

$$\frac{3\times 5\times 8\sqrt{3}\sqrt{55}}{55}$$

Multiply 3 and 5 to get 15.

$$\frac{15\times 8\sqrt{3}\sqrt{55}}{55}$$

Multiply 15 and 8 to get 120.

$$\frac{120\sqrt{3}\sqrt{55}}{55}$$

To multiply \sqrt{3} and \sqrt{55}, multiply the numbers under the square root.

$$\frac{120\sqrt{165}}{55}$$

Divide 120\sqrt{165} by 55 to get \frac{24}{11}\sqrt{165}.

$$\frac{24}{11}\sqrt{165}$$