Question

Suppose that $a$, $b$, and $c$ are real numbers such that $\frac{a}{b} = \frac{\sqrt{10}}{\sqrt{21}}$ and $\frac{b}{c} = \frac{\sqrt{135}}{\sqrt{8}}$. Find $\frac{a}{c}$. Completely simplify and rationalize the denominator.

Answer 1

a/c = (a/b)(b/c) = (√10/√21)(√135/√8).

√135 = 3√15, √8 = 2√2, so
(a/c) = (√10/√21)·(3√15/(2√2)) = (3/2)·(√150/√42) = (3/2)·(5√6/√42) = (15/2)·√(6/42) = (15/2)·√(1/7) = 15/(2√7).

Rationalizing the denominator gives 15√7/14.