Solve [(sqrt{10}-sqrt{15}):sqrt{5}]^2

$$[ ( \sqrt { 10 } - \sqrt { 15 } ) : \sqrt { 5 } ] ^ { 2 }$$

Evaluate

$$5-2\sqrt{6}\approx 0.101020514$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{10}-\sqrt{15}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

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

The square of \sqrt{5} is 5.

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

To raise \frac{\left(\sqrt{10}-\sqrt{15}\right)\sqrt{5}}{5} to a power, raise both numerator and denominator to the power and then divide.

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

Expand \left(\left(\sqrt{10}-\sqrt{15}\right)\sqrt{5}\right)^{2}.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{10}-\sqrt{15}\right)^{2}.

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

The square of \sqrt{10} is 10.

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

To multiply \sqrt{10} and \sqrt{15}, multiply the numbers under the square root.

$$\frac{\left(10-2\sqrt{150}+\left(\sqrt{15}\right)^{2}\right)\left(\sqrt{5}\right)^{2}}{5^{2}}$$

The square of \sqrt{15} is 15.

$$\frac{\left(10-2\sqrt{150}+15\right)\left(\sqrt{5}\right)^{2}}{5^{2}}$$

Add 10 and 15 to get 25.

$$\frac{\left(25-2\sqrt{150}\right)\left(\sqrt{5}\right)^{2}}{5^{2}}$$

Factor 150=5^{2}\times 6. Rewrite the square root of the product \sqrt{5^{2}\times 6} as the product of square roots \sqrt{5^{2}}\sqrt{6}. Take the square root of 5^{2}.

$$\frac{\left(25-2\times 5\sqrt{6}\right)\left(\sqrt{5}\right)^{2}}{5^{2}}$$

Multiply -2 and 5 to get -10.

$$\frac{\left(25-10\sqrt{6}\right)\left(\sqrt{5}\right)^{2}}{5^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(25-10\sqrt{6}\right)\times 5}{5^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{\left(25-10\sqrt{6}\right)\times 5}{25}$$

Divide \left(25-10\sqrt{6}\right)\times 5 by 25 to get \left(25-10\sqrt{6}\right)\times \frac{1}{5}.

$$\left(25-10\sqrt{6}\right)\times \frac{1}{5}$$

Use the distributive property to multiply 25-10\sqrt{6} by \frac{1}{5}.

$$5-2\sqrt{6}$$

Expand

$$5-2\sqrt{6}$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{10}-\sqrt{15}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

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

The square of \sqrt{5} is 5.

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

To raise \frac{\left(\sqrt{10}-\sqrt{15}\right)\sqrt{5}}{5} to a power, raise both numerator and denominator to the power and then divide.

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

Expand \left(\left(\sqrt{10}-\sqrt{15}\right)\sqrt{5}\right)^{2}.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{10}-\sqrt{15}\right)^{2}.

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

The square of \sqrt{10} is 10.

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

To multiply \sqrt{10} and \sqrt{15}, multiply the numbers under the square root.

$$\frac{\left(10-2\sqrt{150}+\left(\sqrt{15}\right)^{2}\right)\left(\sqrt{5}\right)^{2}}{5^{2}}$$

The square of \sqrt{15} is 15.

$$\frac{\left(10-2\sqrt{150}+15\right)\left(\sqrt{5}\right)^{2}}{5^{2}}$$

Add 10 and 15 to get 25.

$$\frac{\left(25-2\sqrt{150}\right)\left(\sqrt{5}\right)^{2}}{5^{2}}$$

Factor 150=5^{2}\times 6. Rewrite the square root of the product \sqrt{5^{2}\times 6} as the product of square roots \sqrt{5^{2}}\sqrt{6}. Take the square root of 5^{2}.

$$\frac{\left(25-2\times 5\sqrt{6}\right)\left(\sqrt{5}\right)^{2}}{5^{2}}$$

Multiply -2 and 5 to get -10.

$$\frac{\left(25-10\sqrt{6}\right)\left(\sqrt{5}\right)^{2}}{5^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(25-10\sqrt{6}\right)\times 5}{5^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{\left(25-10\sqrt{6}\right)\times 5}{25}$$

Divide \left(25-10\sqrt{6}\right)\times 5 by 25 to get \left(25-10\sqrt{6}\right)\times \frac{1}{5}.

$$\left(25-10\sqrt{6}\right)\times \frac{1}{5}$$

Use the distributive property to multiply 25-10\sqrt{6} by \frac{1}{5}.

$$5-2\sqrt{6}$$