Q23. - Algebra 1

Question

Simplify.

${\left(1+\sqrt{2}\right)}^{\mathbf{2}}$

Step-by-Step Solution

Answer

The value of the given expression ${\left(1+\sqrt{2}\right)}^2$ after simplification is $\mathbf{3}\mathbf{+}\mathbf{2}\sqrt{\mathbf{2}}$.

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:

$\begin{array}{c}{\left(1+\sqrt{2}\right)}^2=1^2+2\left(1\right)\left(\sqrt{2}\right)+{\left(\sqrt{2}\right)}^2(\because {(a+b)}^2=a^2+2ab+b^2)\\ =1+2\sqrt{2}+2\\ =1+2+2\sqrt{2}\\ =3+2\sqrt{2}\end{array}$

Step 3. Conclusion

Therefore, the value of the given expression ${\left(1+\sqrt{2}\right)}^2$ after simplification is $3+2\sqrt{2}$.