Writing equivalent rational expressions Convert the first rational expression into an equivalent one that has the indicated denominator. a. 3/2a, ?/6ab b. x – 1/x + 2, ?/x² + 6x + 8 c. a/3b - a, ?/a² - 9b²

Question

Writing equivalent rational expressions

Convert the first rational expression into an equivalent one that has the indicated denominator.

a. [latex] \frac{3}{2 a}, \frac{?}{6 a b}[/latex] b. [latex] \frac{x-1}{x+2}, \frac{?}{x^{2}+6 x+8}[/latex] c. [latex] \frac{a}{3 b-a}, \frac{?}{a^{2}-9 b^{2}}[/latex]

Accepted Answer

a. Compare the two denominators. Since 6ab = 2a(3b), we multiply the numerator and denominator of the first expression by 3b:

[latex] \frac{3}{2 a}=\frac{3 \cdot 3 b}{2 a \cdot 3 b}=\frac{9 b}{6 a b}[/latex]

b. Factor the second denominator and compare it to the first. Since [latex] x^{2}+6 x+8=(x+2)(x+4)[/latex] , we multiply the numerator and denominator by x + 4:

[latex] \frac{x-1}{x+2}=\frac{(x-1)(x+4)}{(x+2)(x+4)}=\frac{x^{2}+3 x-4}{x^{2}+6 x+8}[/latex]

c. Factor the second denominator as

[latex] a^{2}-9 b^{2}=(a-3 b)(a+3 b)=-1(3 b-a)(a+3 b)[/latex]

Since 3b - a is a factor of [latex] a^{2}-9 b^{2}[/latex] , we multiply the numerator and denominator by -1(a + 3b):

[latex] \frac{a}{3 b-a}=\frac{a(-1)(a+3 b)}{(3 b-a)(-1)(a+3 b)}=\frac{-a^{2}-3 a b}{a^{2}-9 b^{2}}[/latex]

Question B.5.5: Writing equivalent rational expressions Convert the first ra...

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