Solve frac{sqrt{33}-sqrt{11}}{sqrt{99}}-frac{sqrt{60}-sqrt{5}}{5sqrt{15}}

$$\frac { \sqrt { 33 } - \sqrt { 11 } } { \sqrt { 99 } } - \frac { \sqrt { 60 } - \sqrt { 5 } } { 5 \sqrt { 15 } }$$

Evaluate

$$\frac{2\sqrt{3}}{5}-\frac{11}{15}\approx -0.04051301$$

Short Solution Steps

Factor 99=3^{2}\times 11. Rewrite the square root of the product \sqrt{3^{2}\times 11} as the product of square roots \sqrt{3^{2}}\sqrt{11}. Take the square root of 3^{2}.

$$\frac{\sqrt{33}-\sqrt{11}}{3\sqrt{11}}-\frac{\sqrt{60}-\sqrt{5}}{5\sqrt{15}}$$

Rationalize the denominator of \frac{\sqrt{33}-\sqrt{11}}{3\sqrt{11}} by multiplying numerator and denominator by \sqrt{11}.

$$\frac{\left(\sqrt{33}-\sqrt{11}\right)\sqrt{11}}{3\left(\sqrt{11}\right)^{2}}-\frac{\sqrt{60}-\sqrt{5}}{5\sqrt{15}}$$

The square of \sqrt{11} is 11.

$$\frac{\left(\sqrt{33}-\sqrt{11}\right)\sqrt{11}}{3\times 11}-\frac{\sqrt{60}-\sqrt{5}}{5\sqrt{15}}$$

Multiply 3 and 11 to get 33.

$$\frac{\left(\sqrt{33}-\sqrt{11}\right)\sqrt{11}}{33}-\frac{\sqrt{60}-\sqrt{5}}{5\sqrt{15}}$$

Factor 60=2^{2}\times 15. Rewrite the square root of the product \sqrt{2^{2}\times 15} as the product of square roots \sqrt{2^{2}}\sqrt{15}. Take the square root of 2^{2}.

$$\frac{\left(\sqrt{33}-\sqrt{11}\right)\sqrt{11}}{33}-\frac{2\sqrt{15}-\sqrt{5}}{5\sqrt{15}}$$

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

$$\frac{\left(\sqrt{33}-\sqrt{11}\right)\sqrt{11}}{33}-\frac{\left(2\sqrt{15}-\sqrt{5}\right)\sqrt{15}}{5\left(\sqrt{15}\right)^{2}}$$

The square of \sqrt{15} is 15.

$$\frac{\left(\sqrt{33}-\sqrt{11}\right)\sqrt{11}}{33}-\frac{\left(2\sqrt{15}-\sqrt{5}\right)\sqrt{15}}{5\times 15}$$

Multiply 5 and 15 to get 75.

$$\frac{\left(\sqrt{33}-\sqrt{11}\right)\sqrt{11}}{33}-\frac{\left(2\sqrt{15}-\sqrt{5}\right)\sqrt{15}}{75}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 33 and 75 is 825. Multiply \frac{\left(\sqrt{33}-\sqrt{11}\right)\sqrt{11}}{33} times \frac{25}{25}. Multiply \frac{\left(2\sqrt{15}-\sqrt{5}\right)\sqrt{15}}{75} times \frac{11}{11}.

$$\frac{25\left(\sqrt{33}-\sqrt{11}\right)\sqrt{11}}{825}-\frac{11\left(2\sqrt{15}-\sqrt{5}\right)\sqrt{15}}{825}$$

Since \frac{25\left(\sqrt{33}-\sqrt{11}\right)\sqrt{11}}{825} and \frac{11\left(2\sqrt{15}-\sqrt{5}\right)\sqrt{15}}{825} have the same denominator, subtract them by subtracting their numerators.

$$\frac{25\left(\sqrt{33}-\sqrt{11}\right)\sqrt{11}-11\left(2\sqrt{15}-\sqrt{5}\right)\sqrt{15}}{825}$$

Do the multiplications in 25\left(\sqrt{33}-\sqrt{11}\right)\sqrt{11}-11\left(2\sqrt{15}-\sqrt{5}\right)\sqrt{15}.

$$\frac{275\sqrt{3}-275-330+55\sqrt{3}}{825}$$

Do the calculations in 275\sqrt{3}-275-330+55\sqrt{3}.

$$\frac{330\sqrt{3}-605}{825}$$

Factor

$$\frac{6 \sqrt{3} - 11}{15} = -0.040513010305782395$$