Question

Simplify

${\left(4-\sqrt{5}\right)}^{\mathbf{2}}$

Short answer

The value of the given expression${\left(4-\sqrt{5}\right)}^2$ after simplification is $\mathbf{21}\mathbf{-}\mathbf{8}\sqrt{\mathbf{5}}$.

Step-by-step solution

Step 1. Write the identity of (a−b)2.

The identity of ${(a-b)}^2$is:

${(a-b)}^2=a^2-2ab+b^2$

Step 2. Simplify the given expression.

The value of the given expression after simplification is:

$\begin{array}{c}{\left(4-\sqrt{5}\right)}^2=4^2-2\left(4\right)\left(\sqrt{5}\right)+{\left(\sqrt{5}\right)}^2(\because {(a-b)}^2=a^2-2ab+b^2)\\ =16-8\sqrt{5}+5\\ =16+5-8\sqrt{5}\\ =21-8\sqrt{5}\end{array}$

Step 3. Conclusion

Therefore, the value of the given expression${\left(4-\sqrt{5}\right)}^2$ after simplification is $21-8\sqrt{5}$.