Solve frac{sqrt{27}-sqrt{15}}{sqrt{2}}*frac{sqrt{3}}{sqrt{10}}

$$\frac { \sqrt { 27 } - \sqrt { 15 } } { \sqrt { 2 } } \times \frac { \sqrt { 3 } } { \sqrt { 10 } }$$

Evaluate

$$\frac{9\sqrt{5}}{10}-\frac{3}{2}\approx 0.51246118$$

Solution Steps

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

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

Rationalize the denominator of \frac{3\sqrt{3}-\sqrt{15}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

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

Rationalize the denominator of \frac{\sqrt{3}}{\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

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

The square of \sqrt{10} is 10.

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

To multiply \sqrt{3} and \sqrt{10}, multiply the numbers under the square root.

$$\frac{\left(3\sqrt{3}-\sqrt{15}\right)\sqrt{2}}{2}\times \frac{\sqrt{30}}{10}$$

Multiply \frac{\left(3\sqrt{3}-\sqrt{15}\right)\sqrt{2}}{2} times \frac{\sqrt{30}}{10} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(3\sqrt{3}-\sqrt{15}\right)\sqrt{2}\sqrt{30}}{2\times 10}$$

Factor 30=2\times 15. Rewrite the square root of the product \sqrt{2\times 15} as the product of square roots \sqrt{2}\sqrt{15}.

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

Multiply \sqrt{2} and \sqrt{2} to get 2.

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

Multiply 2 and 10 to get 20.

$$\frac{\left(3\sqrt{3}-\sqrt{15}\right)\times 2\sqrt{15}}{20}$$

Divide \left(3\sqrt{3}-\sqrt{15}\right)\times 2\sqrt{15} by 20 to get \left(3\sqrt{3}-\sqrt{15}\right)\times \frac{1}{10}\sqrt{15}.

$$\left(3\sqrt{3}-\sqrt{15}\right)\times \frac{1}{10}\sqrt{15}$$

Use the distributive property to multiply 3\sqrt{3}-\sqrt{15} by \frac{1}{10}.

$$\left(3\sqrt{3}\times \frac{1}{10}-\sqrt{15}\times \frac{1}{10}\right)\sqrt{15}$$

Multiply 3 and \frac{1}{10} to get \frac{3}{10}.

$$\left(\frac{3}{10}\sqrt{3}-\sqrt{15}\times \frac{1}{10}\right)\sqrt{15}$$

Multiply -1 and \frac{1}{10} to get -\frac{1}{10}.

$$\left(\frac{3}{10}\sqrt{3}-\frac{1}{10}\sqrt{15}\right)\sqrt{15}$$

Use the distributive property to multiply \frac{3}{10}\sqrt{3}-\frac{1}{10}\sqrt{15} by \sqrt{15}.

$$\frac{3}{10}\sqrt{3}\sqrt{15}-\frac{1}{10}\sqrt{15}\sqrt{15}$$

Multiply \sqrt{15} and \sqrt{15} to get 15.

$$\frac{3}{10}\sqrt{3}\sqrt{15}-\frac{1}{10}\times 15$$

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

$$\frac{3}{10}\sqrt{3}\sqrt{3}\sqrt{5}-\frac{1}{10}\times 15$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{3}{10}\times 3\sqrt{5}-\frac{1}{10}\times 15$$

Express \frac{3}{10}\times 3 as a single fraction.

$$\frac{3\times 3}{10}\sqrt{5}-\frac{1}{10}\times 15$$

Multiply 3 and 3 to get 9.

$$\frac{9}{10}\sqrt{5}-\frac{1}{10}\times 15$$

Express -\frac{1}{10}\times 15 as a single fraction.

$$\frac{9}{10}\sqrt{5}+\frac{-15}{10}$$

Reduce the fraction \frac{-15}{10} to lowest terms by extracting and canceling out 5.

$$\frac{9}{10}\sqrt{5}-\frac{3}{2}$$