Solve frac{sqrt{33}-sqrt{11}}{sqrt{99}}frac{sqrt{60}-sqrt{50}}{5sqrt{15}}

$$\frac { \sqrt { 33 } - \sqrt { 11 } } { \sqrt { 99 } } \frac { \sqrt { 60 } - \sqrt { 50 } } { 5 \sqrt { 15 } }$$

Evaluate

$$\frac{\sqrt{30}+6\sqrt{3}-3\sqrt{10}-6}{45}\approx 0.008504388$$

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}.

$$\frac{\sqrt{33}-\sqrt{11}}{3\sqrt{11}}\times \frac{\sqrt{60}-\sqrt{50}}{5\sqrt{15}}$$

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

$$\frac{\left(\sqrt{33}-\sqrt{11}\right)\sqrt{11}}{3\left(\sqrt{11}\right)^{2}}\times \frac{\sqrt{60}-\sqrt{50}}{5\sqrt{15}}$$

The square of \sqrt{11} is 11.

$$\frac{\left(\sqrt{33}-\sqrt{11}\right)\sqrt{11}}{3\times 11}\times \frac{\sqrt{60}-\sqrt{50}}{5\sqrt{15}}$$

Multiply 3 and 11 to get 33.

$$\frac{\left(\sqrt{33}-\sqrt{11}\right)\sqrt{11}}{33}\times \frac{\sqrt{60}-\sqrt{50}}{5\sqrt{15}}$$

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}.

$$\frac{\left(\sqrt{33}-\sqrt{11}\right)\sqrt{11}}{33}\times \frac{2\sqrt{15}-\sqrt{50}}{5\sqrt{15}}$$

Factor 50=5^{2}\times 2. Rewrite the square root of the product \sqrt{5^{2}\times 2} as the product of square roots \sqrt{5^{2}}\sqrt{2}. Take the square root of 5^{2}.

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

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

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

The square of \sqrt{15} is 15.

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

Multiply 5 and 15 to get 75.

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

Multiply \frac{\left(\sqrt{33}-\sqrt{11}\right)\sqrt{11}}{33} times \frac{\left(2\sqrt{15}-5\sqrt{2}\right)\sqrt{15}}{75} by multiplying numerator times numerator and denominator times denominator.

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

To multiply \sqrt{11} and \sqrt{15}, multiply the numbers under the square root.

$$\frac{\left(\sqrt{33}-\sqrt{11}\right)\sqrt{165}\left(2\sqrt{15}-5\sqrt{2}\right)}{33\times 75}$$

Multiply 33 and 75 to get 2475.

$$\frac{\left(\sqrt{33}-\sqrt{11}\right)\sqrt{165}\left(2\sqrt{15}-5\sqrt{2}\right)}{2475}$$

Use the distributive property to multiply \sqrt{33}-\sqrt{11} by \sqrt{165}.

$$\frac{\left(\sqrt{33}\sqrt{165}-\sqrt{11}\sqrt{165}\right)\left(2\sqrt{15}-5\sqrt{2}\right)}{2475}$$

Factor 165=33\times 5. Rewrite the square root of the product \sqrt{33\times 5} as the product of square roots \sqrt{33}\sqrt{5}.

$$\frac{\left(\sqrt{33}\sqrt{33}\sqrt{5}-\sqrt{11}\sqrt{165}\right)\left(2\sqrt{15}-5\sqrt{2}\right)}{2475}$$

Multiply \sqrt{33} and \sqrt{33} to get 33.

$$\frac{\left(33\sqrt{5}-\sqrt{11}\sqrt{165}\right)\left(2\sqrt{15}-5\sqrt{2}\right)}{2475}$$

Factor 165=11\times 15. Rewrite the square root of the product \sqrt{11\times 15} as the product of square roots \sqrt{11}\sqrt{15}.

$$\frac{\left(33\sqrt{5}-\sqrt{11}\sqrt{11}\sqrt{15}\right)\left(2\sqrt{15}-5\sqrt{2}\right)}{2475}$$

Multiply \sqrt{11} and \sqrt{11} to get 11.

$$\frac{\left(33\sqrt{5}-11\sqrt{15}\right)\left(2\sqrt{15}-5\sqrt{2}\right)}{2475}$$

Apply the distributive property by multiplying each term of 33\sqrt{5}-11\sqrt{15} by each term of 2\sqrt{15}-5\sqrt{2}.

$$\frac{66\sqrt{5}\sqrt{15}-165\sqrt{5}\sqrt{2}-22\left(\sqrt{15}\right)^{2}+55\sqrt{15}\sqrt{2}}{2475}$$

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}.

$$\frac{66\sqrt{5}\sqrt{5}\sqrt{3}-165\sqrt{5}\sqrt{2}-22\left(\sqrt{15}\right)^{2}+55\sqrt{15}\sqrt{2}}{2475}$$

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

$$\frac{66\times 5\sqrt{3}-165\sqrt{5}\sqrt{2}-22\left(\sqrt{15}\right)^{2}+55\sqrt{15}\sqrt{2}}{2475}$$

Multiply 66 and 5 to get 330.

$$\frac{330\sqrt{3}-165\sqrt{5}\sqrt{2}-22\left(\sqrt{15}\right)^{2}+55\sqrt{15}\sqrt{2}}{2475}$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{330\sqrt{3}-165\sqrt{10}-22\left(\sqrt{15}\right)^{2}+55\sqrt{15}\sqrt{2}}{2475}$$

The square of \sqrt{15} is 15.

$$\frac{330\sqrt{3}-165\sqrt{10}-22\times 15+55\sqrt{15}\sqrt{2}}{2475}$$

Multiply -22 and 15 to get -330.

$$\frac{330\sqrt{3}-165\sqrt{10}-330+55\sqrt{15}\sqrt{2}}{2475}$$

To multiply \sqrt{15} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{330\sqrt{3}-165\sqrt{10}-330+55\sqrt{30}}{2475}$$