(5 + square root 65)(5 + square root 143)

Answer 1

To simplify the expression \((5 + \sqrt{65})(5 + \sqrt{143})\), we can use the distributive property (also known as the FOIL method for binomials).

First, we expand the expression:

\[ (5 + \sqrt{65})(5 + \sqrt{143}) = 5 \times 5 + 5 \times \sqrt{143} + \sqrt{65} \times 5 + \sqrt{65} \times \sqrt{143} \]

This can be simplified as follows:

1. \(5 \times 5 = 25\)
2. \(5 \times \sqrt{143} = 5\sqrt{143}\)
3. \(\sqrt{65} \times 5 = 5\sqrt{65}\)
4. \(\sqrt{65} \times \sqrt{143} = \sqrt{65 \times 143}\)

Now, calculate \(65 \times 143\):

\[ 65 \times 143 = 9295 \]

Thus, \(\sqrt{65 \times 143} = \sqrt{9295}\).

Putting it all together, we have:

\[ (5 + \sqrt{65})(5 + \sqrt{143}) = 25 + 5\sqrt{143} + 5\sqrt{65} + \sqrt{9295} \]

So the final answer is:

\[ 25 + 5\sqrt{143} + 5\sqrt{65} + \sqrt{9295} \]