Question

Simplify $(3-2\sqrt{5})(3+2\sqrt{5})$.

Short answer

The value is$-11$.

Step-by-step solution

Step 1. Given and explanation

We have $(3-2\sqrt{5})(3+2\sqrt{5})$.

We can identify the property of product of conjugates, i.e., $(a+b)(a-b)$.

We know that $(a+b)(a-b)={\left(a\right)}^2-{\left(b\right)}^2$.

By applying it, we will get the value.

Step 2. Simplifying and evaluating value.

We have $(3-2\sqrt{5})(3+2\sqrt{5})$.

Applying the property $(a+b)(a-b)$will give us,

role="math" localid="1644766740151" $$\begin{gathered} =(3-2\sqrt{5})(3+2\sqrt{5}) \\ ={\left(3\right)}^2-{\left(2\sqrt{5}\right)}^2 \\ =9-20 \\ =-11 \end{gathered}$$

Thus we get the value as$-11$.