How to Answer? (Surd Qu's): Express (√5 - 3√2)/(√5 -√2) as the sum of two numbers with rational denominators.

If someone could walk me through this with the working out included, it would be greatly appreciated. Thanks!

Multiply the numerator and denominator by the conjugate of the denominator. The conjugate of √5 - √2 is √5 + √2 (just take the minus sign and make it positive), so multiplying the numerator and denominator by √5 + √2 gives:
(√5 - 3√2)/(√5 - √2) = [(√5 - 3√2)(√5 + √2)]/[(√5 + √2)(√5 - √2)]
= [(√5 - 3√2)(√5 + √2)]/[(√5)^2 - (√2)^2], via difference of squares
= [(√5 - 3√2)(√5 + √2)]/(5 - 2)
= [(√5 - 3√2)(√5 + √2)]/3
= (√5√5 + √2√5 - 3√2√5 - 3√2√2)/3, by FOILing the numerator
= [5 + √10 - 3√10 - 3(2)]/3
= (-1 - 2√10)/3.

I hope this helps!

Multiply the numerator and denominator by sqrt(5) + sqrt(2) because:

(a - b) * (a + b) = a^2 - b^2

((sqrt(5) - 3 * sqrt(2)) * (sqrt(5) + sqrt(2))) / ((sqrt(5) - sqrt(2)) * (sqrt(5) + sqrt(2))) =>
(sqrt(5) * sqrt(5) - 3 * sqrt(2) * sqrt(5) + sqrt(5) * sqrt(2) - 3 * sqrt(2) * sqrt(2)) / (5 - 2) =>
(5 - 3 * sqrt(10) + sqrt(10) - 3 * 2) / 3 =>
(5 - 6 - 2 * sqrt(10)) / 3 =>
(-1 - 2 * sqrt(10)) / 3

There you go.