Solve (sqrt{5}-sqrt{6})^2+sqrt{81}+2sqrt{30}

$$( \sqrt { 5 } - \sqrt { 6 } ) ^ { 2 } + \sqrt { 81 } + 2 \sqrt { 30 }$$

Evaluate

$$20$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{5}-\sqrt{6}\right)^{2}.

$$\left(\sqrt{5}\right)^{2}-2\sqrt{5}\sqrt{6}+\left(\sqrt{6}\right)^{2}+\sqrt{81}+2\sqrt{30}$$

The square of \sqrt{5} is 5.

$$5-2\sqrt{5}\sqrt{6}+\left(\sqrt{6}\right)^{2}+\sqrt{81}+2\sqrt{30}$$

To multiply \sqrt{5} and \sqrt{6}, multiply the numbers under the square root.

$$5-2\sqrt{30}+\left(\sqrt{6}\right)^{2}+\sqrt{81}+2\sqrt{30}$$

The square of \sqrt{6} is 6.

$$5-2\sqrt{30}+6+\sqrt{81}+2\sqrt{30}$$

Add 5 and 6 to get 11.

$$11-2\sqrt{30}+\sqrt{81}+2\sqrt{30}$$

Calculate the square root of 81 and get 9.

$$11-2\sqrt{30}+9+2\sqrt{30}$$

Add 11 and 9 to get 20.

$$20-2\sqrt{30}+2\sqrt{30}$$

Combine -2\sqrt{30} and 2\sqrt{30} to get 0.

$$20$$

Factor

$$2^{2}\times 5$$