Q. 8.91 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The value is $-11$.

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,

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

Thus we get the value as $-11$.