Solve {left(sqrt{5}-sqrt{2}right)}^265

$${ \left( \sqrt{ 5 } - \sqrt{ 2 } \right) }^{ 2 } 65$$

Evaluate

$$455-130\sqrt{10}\approx 43.903904178$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{5}-\sqrt{2}\right)^{2}.

$$65\left(\left(\sqrt{5}\right)^{2}-2\sqrt{5}\sqrt{2}+\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{5} is 5.

$$65\left(5-2\sqrt{5}\sqrt{2}+\left(\sqrt{2}\right)^{2}\right)$$

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

$$65\left(5-2\sqrt{10}+\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{2} is 2.

$$65\left(5-2\sqrt{10}+2\right)$$

Add 5 and 2 to get 7.

$$65\left(7-2\sqrt{10}\right)$$

Use the distributive property to multiply 65 by 7-2\sqrt{10}.

$$455-130\sqrt{10}$$

Expand

$$455-130\sqrt{10}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{5}-\sqrt{2}\right)^{2}.

$$65\left(\left(\sqrt{5}\right)^{2}-2\sqrt{5}\sqrt{2}+\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{5} is 5.

$$65\left(5-2\sqrt{5}\sqrt{2}+\left(\sqrt{2}\right)^{2}\right)$$

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

$$65\left(5-2\sqrt{10}+\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{2} is 2.

$$65\left(5-2\sqrt{10}+2\right)$$

Add 5 and 2 to get 7.

$$65\left(7-2\sqrt{10}\right)$$

Use the distributive property to multiply 65 by 7-2\sqrt{10}.

$$455-130\sqrt{10}$$