Q. 8.29 - Intermediate Algebra

Question

Simplify
$(4+5\sqrt{7})(4-5\sqrt{7})$.

Step-by-Step Solution

Answer

The value is $-159$.

Step 1. Given and explanation

We have $(4+5\sqrt{7})(4-5\sqrt{7})$.

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 $(4+5\sqrt{7})(4-5\sqrt{7})$.

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

$$\begin{gathered} =(4+5\sqrt{7})(4-5\sqrt{7}) \\ ={\left(4\right)}^2-{\left(5\sqrt{7}\right)}^2 \\ =16-175 \\ =-159 \end{gathered}$$

Thus we get the value as $-159$.