Solve 2sqrt{24}div3sqrt{3/5}*1/3sqrt{24/5}

$$2 \sqrt { 24 } \div 3 \sqrt { \frac { 3 } { 5 } } \times \frac { 1 } { 3 } \sqrt { 2 \frac { 4 } { 5 } }$$

Evaluate

$$\frac{8\sqrt{7}}{15}\approx 1.411067366$$

Solution Steps

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}.

$$\frac{2\times 2\sqrt{6}}{3}\sqrt{\frac{3}{5}}\times \frac{1}{3}\sqrt{\frac{2\times 5+4}{5}}$$

Multiply 2 and 2 to get 4.

$$\frac{4\sqrt{6}}{3}\sqrt{\frac{3}{5}}\times \frac{1}{3}\sqrt{\frac{2\times 5+4}{5}}$$

Rewrite the square root of the division \sqrt{\frac{3}{5}} as the division of square roots \frac{\sqrt{3}}{\sqrt{5}}.

$$\frac{4\sqrt{6}}{3}\times \frac{\sqrt{3}}{\sqrt{5}}\times \frac{1}{3}\sqrt{\frac{2\times 5+4}{5}}$$

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

$$\frac{4\sqrt{6}}{3}\times \frac{\sqrt{3}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}\times \frac{1}{3}\sqrt{\frac{2\times 5+4}{5}}$$

The square of \sqrt{5} is 5.

$$\frac{4\sqrt{6}}{3}\times \frac{\sqrt{3}\sqrt{5}}{5}\times \frac{1}{3}\sqrt{\frac{2\times 5+4}{5}}$$

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

$$\frac{4\sqrt{6}}{3}\times \frac{\sqrt{15}}{5}\times \frac{1}{3}\sqrt{\frac{2\times 5+4}{5}}$$

Multiply 2 and 5 to get 10.

$$\frac{4\sqrt{6}}{3}\times \frac{\sqrt{15}}{5}\times \frac{1}{3}\sqrt{\frac{10+4}{5}}$$

Add 10 and 4 to get 14.

$$\frac{4\sqrt{6}}{3}\times \frac{\sqrt{15}}{5}\times \frac{1}{3}\sqrt{\frac{14}{5}}$$

Rewrite the square root of the division \sqrt{\frac{14}{5}} as the division of square roots \frac{\sqrt{14}}{\sqrt{5}}.

$$\frac{4\sqrt{6}}{3}\times \frac{\sqrt{15}}{5}\times \frac{1}{3}\times \frac{\sqrt{14}}{\sqrt{5}}$$

Rationalize the denominator of \frac{\sqrt{14}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{4\sqrt{6}}{3}\times \frac{\sqrt{15}}{5}\times \frac{1}{3}\times \frac{\sqrt{14}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{4\sqrt{6}}{3}\times \frac{\sqrt{15}}{5}\times \frac{1}{3}\times \frac{\sqrt{14}\sqrt{5}}{5}$$

To multiply \sqrt{14} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{4\sqrt{6}}{3}\times \frac{\sqrt{15}}{5}\times \frac{1}{3}\times \frac{\sqrt{70}}{5}$$

Multiply \frac{4\sqrt{6}}{3} times \frac{\sqrt{15}}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{4\sqrt{6}\sqrt{15}}{3\times 5}\times \frac{1}{3}\times \frac{\sqrt{70}}{5}$$

Multiply \frac{4\sqrt{6}\sqrt{15}}{3\times 5} times \frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{4\sqrt{6}\sqrt{15}}{3\times 5\times 3}\times \frac{\sqrt{70}}{5}$$

Multiply \frac{4\sqrt{6}\sqrt{15}}{3\times 5\times 3} times \frac{\sqrt{70}}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{4\sqrt{6}\sqrt{15}\sqrt{70}}{3\times 5\times 3\times 5}$$

To multiply \sqrt{6} and \sqrt{15}, multiply the numbers under the square root.

$$\frac{4\sqrt{90}\sqrt{70}}{3\times 5\times 3\times 5}$$

To multiply \sqrt{90} and \sqrt{70}, multiply the numbers under the square root.

$$\frac{4\sqrt{6300}}{3\times 5\times 3\times 5}$$

Multiply 3 and 5 to get 15.

$$\frac{4\sqrt{6300}}{15\times 3\times 5}$$

Multiply 15 and 3 to get 45.

$$\frac{4\sqrt{6300}}{45\times 5}$$

Multiply 45 and 5 to get 225.

$$\frac{4\sqrt{6300}}{225}$$

Factor 6300=30^{2}\times 7. Rewrite the square root of the product \sqrt{30^{2}\times 7} as the product of square roots \sqrt{30^{2}}\sqrt{7}. Take the square root of 30^{2}.

$$\frac{4\times 30\sqrt{7}}{225}$$

Multiply 4 and 30 to get 120.

$$\frac{120\sqrt{7}}{225}$$

Divide 120\sqrt{7} by 225 to get \frac{8}{15}\sqrt{7}.

$$\frac{8}{15}\sqrt{7}$$