Solve =frac{(sqrt{27}-sqrt{15})(sqrt{3})}{sqrt{20}}

$$= \frac { ( \sqrt { 27 } - \sqrt { 15 } ) ( \sqrt { 3 } ) } { \sqrt { 20 } }$$

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}.

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

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$\frac{\left(3\sqrt{3}-\sqrt{15}\right)\sqrt{3}}{2\sqrt{5}}$$

Rationalize the denominator of \frac{\left(3\sqrt{3}-\sqrt{15}\right)\sqrt{3}}{2\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

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

The square of \sqrt{5} is 5.

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

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

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

Multiply 2 and 5 to get 10.

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

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

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

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}.

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

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

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

Multiply 3 and 3 to get 9.

$$\frac{9\sqrt{5}-\left(\sqrt{15}\right)^{2}}{10}$$

The square of \sqrt{15} is 15.

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