How do you rationalize the denominator and simplify (sqrtc-sqrtd)/(sqrtc+sqrtd)?

1 Answer
George C.
Oct 3, 2015

Multiply numerator and denominator by (sqrt(c)-sqrt(d)) and simplify to find:

(sqrt(c)-sqrt(d))/(sqrt(c)+sqrt(d)) = (c+d-2sqrt(cd))/(c-d)

Explanation:

Use the difference of squares identity: a^2-b^2 = (a-b)(a+b) with a = sqrt(c) and b=sqrt(d)

Also use sqrt(a)sqrt(b) = sqrt(ab) (if a, b >= 0)

(sqrt(c)-sqrt(d))/(sqrt(c)+sqrt(d))

= ((sqrt(c)-sqrt(d))(sqrt(c)-sqrt(d)))/((sqrt(c)-sqrt(d))(sqrt(c)+sqrt(d)))

= ((sqrt(c))^2-2(sqrt(c))(sqrt(d))+(sqrt(d))^2)/((sqrt(c))^2 - (sqrt(d))^2)

= (c+d-2sqrt(cd))/(c-d)

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