Question

Evaluate:

`(sqrt(31) + sqrt(56))^(1/2) (sqrt(56) - sqrt(31))^(1/2)`

Answer 1

We are asked to evaluate:

`(sqrt(31) + sqrt(56))^(1/2) * (sqrt(56) - sqrt(31))^(1/2)`

Step 1: Use identity

`sqrt(a + b) * sqrt(b - a) = sqrt((a + b)(b - a))` if `a = sqrt(31), b = sqrt(56)`

So, `(sqrt(31) + sqrt(56))^(1/2) * (sqrt(56) - sqrt(31))^(1/2) = [(sqrt(56) + sqrt(31))(sqrt(56) - sqrt(31))]^(1/2)`

Step 2: Apply difference of squares

`(sqrt(56) + sqrt(31))(sqrt(56) - sqrt(31))`

= 56 – 31

= 25

Step 3: Take square root

`sqrt(25)` = \[\boxed{5}\]