Question

5-70. rewrite each of the following expressions as a single fraction without negative exponents. homework help a. \\(\frac{3x^{-3} + 4x}{2x - x^{-2}}\\) b. \\(\frac{x^{-2} + y^{2}}{x^{2} - y^{-2}}\\) c. \\(\frac{x^{-4}y^{-2} + x^{-3}y^{-1}}{x^{-1} - x^{-3}}\\)

Explanation:

Part a

Step1: Eliminate negative exponents

Multiply numerator and denominator by $x^3$:
$$\frac{(3x^{-3}+4x) \cdot x^3}{(2x-x^{-2}) \cdot x^3} = \frac{3 + 4x^4}{2x^4 - x}$$

Step2: Factor denominator

Factor out $x$ from denominator:
$$\frac{3 + 4x^4}{x(2x^3 - 1)}$$

Part b

Step1: Eliminate negative exponents

Multiply numerator and denominator by $x^2y^2$:
$$\frac{(x^{-2}+y^2) \cdot x^2y^2}{(x^2-y^{-2}) \cdot x^2y^2} = \frac{y^2 + x^2y^4}{x^4y^2 - x^2}$$

Step2: Factor numerator/denominator

Factor numerator: $y^2(1 + x^2y^2)$; Factor denominator: $x^2(x^2y^2 - 1)$
$$\frac{y^2(1 + x^2y^2)}{x^2(x^2y^2 - 1)}$$

Part c

Step1: Eliminate negative exponents

Multiply numerator and denominator by $x^4y^2$:
$$\frac{(x^{-4}y^{-2}+x^{-3}y^{-1}) \cdot x^4y^2}{(x^{-1}-x^{-3}) \cdot x^4y^2} = \frac{1 + xy}{x^3y^2 - xy^2}$$

Step2: Factor numerator/denominator

Factor denominator: $xy^2(x^2 - 1) = xy^2(x-1)(x+1)$
$$\frac{1 + xy}{xy^2(x-1)(x+1)}$$

Answer:

a. $\boldsymbol{\frac{4x^4 + 3}{x(2x^3 - 1)}}$
b. $\boldsymbol{\frac{y^2(x^2y^2 + 1)}{x^2(x^2y^2 - 1)}}$
c. $\boldsymbol{\frac{xy + 1}{xy^2(x-1)(x+1)}}$