Solve sqrt{(frac{2sqrt{6}}{3})^2-(frac{2sqrt{15}}{15})^2}

$$\sqrt { ( \frac { 2 \sqrt { 6 } } { 3 } ) ^ { 2 } - ( \frac { 2 \sqrt { 15 } } { 15 } ) ^ { 2 } }$$

Evaluate

$$\frac{2\sqrt{15}}{5}\approx 1.549193338$$

Solution Steps

To raise \frac{2\sqrt{6}}{3} to a power, raise both numerator and denominator to the power and then divide.

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

To raise \frac{2\sqrt{15}}{15} to a power, raise both numerator and denominator to the power and then divide.

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

Expand \left(2\sqrt{15}\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

$$\sqrt{\frac{\left(2\sqrt{6}\right)^{2}}{3^{2}}-\frac{4\left(\sqrt{15}\right)^{2}}{15^{2}}}$$

The square of \sqrt{15} is 15.

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

Multiply 4 and 15 to get 60.

$$\sqrt{\frac{\left(2\sqrt{6}\right)^{2}}{3^{2}}-\frac{60}{15^{2}}}$$

Calculate 15 to the power of 2 and get 225.

$$\sqrt{\frac{\left(2\sqrt{6}\right)^{2}}{3^{2}}-\frac{60}{225}}$$

Reduce the fraction \frac{60}{225} to lowest terms by extracting and canceling out 15.

$$\sqrt{\frac{\left(2\sqrt{6}\right)^{2}}{3^{2}}-\frac{4}{15}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3^{2} and 15 is 45. Multiply \frac{\left(2\sqrt{6}\right)^{2}}{3^{2}} times \frac{5}{5}. Multiply \frac{4}{15} times \frac{3}{3}.

$$\sqrt{\frac{5\times \left(2\sqrt{6}\right)^{2}}{45}-\frac{4\times 3}{45}}$$

Since \frac{5\times \left(2\sqrt{6}\right)^{2}}{45} and \frac{4\times 3}{45} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{5\times \left(2\sqrt{6}\right)^{2}-4\times 3}{45}}$$

Expand \left(2\sqrt{6}\right)^{2}.

$$\sqrt{\frac{5\times 2^{2}\left(\sqrt{6}\right)^{2}-4\times 3}{45}}$$

Calculate 2 to the power of 2 and get 4.

$$\sqrt{\frac{5\times 4\left(\sqrt{6}\right)^{2}-4\times 3}{45}}$$

The square of \sqrt{6} is 6.

$$\sqrt{\frac{5\times 4\times 6-4\times 3}{45}}$$

Multiply 4 and 6 to get 24.

$$\sqrt{\frac{5\times 24-4\times 3}{45}}$$

Multiply 5 and 24 to get 120.

$$\sqrt{\frac{120-4\times 3}{45}}$$

Multiply -4 and 3 to get -12.

$$\sqrt{\frac{120-12}{45}}$$

Subtract 12 from 120 to get 108.

$$\sqrt{\frac{108}{45}}$$

Reduce the fraction \frac{108}{45} to lowest terms by extracting and canceling out 9.

$$\sqrt{\frac{12}{5}}$$

Rewrite the square root of the division \sqrt{\frac{12}{5}} as the division of square roots \frac{\sqrt{12}}{\sqrt{5}}.

$$\frac{\sqrt{12}}{\sqrt{5}}$$

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

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

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

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

The square of \sqrt{5} is 5.

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

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

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