Question

Simplify:

`(sqrt(6) + sqrt(10))/(sqrt(27) + sqrt(45))`

Answer 1

`(sqrt(6) + sqrt(10))/(sqrt(27) + sqrt(45))`

Step 1: Simplify each square root

• `sqrt(27) = sqrt(9 xx 3) = 3sqrt(3)`
• `sqrt(45) = sqrt(9 xx 5) = 3sqrt(5)`

So the denominator becomes `3sqrt(3) + 3sqrt(5) = 3(sqrt(3) + sqrt(5))`.

Step 2: Rewrite numerator

`sqrt(6) + sqrt(10)` cannot be simplified further into integers, so we leave it as is.

So the expression becomes `(sqrt(6) + sqrt(10))/(3(sqrt(3) + sqrt(5))`.

Step 3: Factor numerator cleverly

Notice: `sqrt(6) + sqrt(10)`

= `sqrt(2 xx 3) + sqrt(2 xx 5)`

= `sqrt(2)(sqrt(3) + sqrt(5))`

So numerator = `sqrt(2)(sqrt(3) + sqrt(5))`.

Step 4: Cancel common factor

`(sqrt(2)(sqrt(3) + sqrt(5)))/(3(sqrt(3) + sqrt(5))`

Cancel `(sqrt(3) + sqrt(5)) = sqrt(2)/3`.