Question

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

Short answer

The value is$-159$.

Step-by-step solution

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,

role="math" localid="1644767098828" $$\begin{gathered} =(4+5\sqrt{7})(4-5\sqrt{7}) \\ ={\left(4\right)}^2-{\left(5\sqrt{7}\right)}^2 \\ =16-175 \\ =-159 \end{gathered}$$

Thus we get the value as$-159$.