Solve 6sqrt{frac{5/3}-10sqrt{3/5}+2sqrt{15}}{sqrt{5}}

$$\frac { 6 \sqrt { \frac { 5 } { 3 } } - 10 \sqrt { \frac { 3 } { 5 } } + 2 \sqrt { 15 } } { \sqrt { 5 } }$$

Evaluate

$$2\sqrt{3}\approx 3.464101615$$

Solution Steps

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

$$\frac{6\times \frac{\sqrt{5}}{\sqrt{3}}-10\sqrt{\frac{3}{5}}+2\sqrt{15}}{\sqrt{5}}$$

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

$$\frac{6\times \frac{\sqrt{5}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-10\sqrt{\frac{3}{5}}+2\sqrt{15}}{\sqrt{5}}$$

The square of \sqrt{3} is 3.

$$\frac{6\times \frac{\sqrt{5}\sqrt{3}}{3}-10\sqrt{\frac{3}{5}}+2\sqrt{15}}{\sqrt{5}}$$

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

$$\frac{6\times \frac{\sqrt{15}}{3}-10\sqrt{\frac{3}{5}}+2\sqrt{15}}{\sqrt{5}}$$

Cancel out 3, the greatest common factor in 6 and 3.

$$\frac{2\sqrt{15}-10\sqrt{\frac{3}{5}}+2\sqrt{15}}{\sqrt{5}}$$

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

$$\frac{2\sqrt{15}-10\times \frac{\sqrt{3}}{\sqrt{5}}+2\sqrt{15}}{\sqrt{5}}$$

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

$$\frac{2\sqrt{15}-10\times \frac{\sqrt{3}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}+2\sqrt{15}}{\sqrt{5}}$$

The square of \sqrt{5} is 5.

$$\frac{2\sqrt{15}-10\times \frac{\sqrt{3}\sqrt{5}}{5}+2\sqrt{15}}{\sqrt{5}}$$

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

$$\frac{2\sqrt{15}-10\times \frac{\sqrt{15}}{5}+2\sqrt{15}}{\sqrt{5}}$$

Cancel out 5, the greatest common factor in 10 and 5.

$$\frac{2\sqrt{15}-2\sqrt{15}+2\sqrt{15}}{\sqrt{5}}$$

Combine 2\sqrt{15} and -2\sqrt{15} to get 0.

$$\frac{2\sqrt{15}}{\sqrt{5}}$$

Rationalize the denominator of \frac{2\sqrt{15}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{2\sqrt{15}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{2\sqrt{15}\sqrt{5}}{5}$$

Factor 15=5\times 3. Rewrite the square root of the product \sqrt{5\times 3} as the product of square roots \sqrt{5}\sqrt{3}.

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

Multiply \sqrt{5} and \sqrt{5} to get 5.

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

Cancel out 5 and 5.

$$2\sqrt{3}$$