How do you simplify (y – x)/(x^2y) + (x + y) (xy^2)?

Apr 21, 2018

See explanation

Explanation:

Rules used:
Negative exponents $\frac{1}{x} ^ a = {x}^{-} a$
Exponent of exponent value ${\left({a}^{b}\right)}^{c} = {a}^{b \times c}$
Exponential multiplication ${a}^{b} \cdot {a}^{c} = {a}^{b + c}$
Distributive property $a \left(b + c\right) = a \left(b\right) + a \left(c\right)$
Zero exponent ${z}^{0} = 1$

Steps:
1. Simplify the first half.
Negative exponents $\frac{y - x}{{x}^{2} y} = \left(y - x\right) \cdot {\left({x}^{2} y\right)}^{-} 1$
Exponent of exponent value $y \left({x}^{-} 2 {y}^{-} 1\right) - x \left({x}^{-} 2 {y}^{-} 1\right)$
Distributive property $\left(y - x\right) \left({x}^{-} 2 {y}^{-} 1\right) = {\left({x}^{2} y\right)}^{-} 1 = {x}^{2 \times \left(- 1\right)} {y}^{-} 1 = {x}^{-} 2 {y}^{-} 1$
Exponential multiplication ${y}^{1 - 1} {x}^{-} 2 - {x}^{1 - 2} {y}^{-} 1 = {y}^{0} {x}^{-} 2 - {x}^{-} 1 {y}^{-} 1$
Zero exponent $\left(1\right) {x}^{-} 2 - {x}^{-} 1 {y}^{-} 1$
Negative exponent $\frac{1}{x} ^ 2 - \frac{1}{x y}$
Result: $\setminus \textcolor{\to m a \to}{\frac{1}{x} ^ 2 - \frac{1}{x y}}$
2. Simplify the second half.
Distributive property $\left(x + y\right) \left(x {y}^{2}\right) = x \left(x {y}^{2}\right) + y \left(x {y}^{2}\right)$
Exponential multiplication $x \left(x {y}^{2}\right) = {x}^{1 + 1} {y}^{2} = {x}^{2} {y}^{2}$ and $y \left(x {y}^{2}\right) = x {y}^{1 + 2} = x {y}^{3}$
Result: $\setminus \textcolor{\mathmr{and} \chi d}{{x}^{2} {y}^{2} + x {y}^{3}}$
3. Combine these simplified halves.
$\setminus \textcolor{\to m a \to}{\frac{1}{x} ^ 2 - \frac{1}{x y}} + \setminus \textcolor{\mathmr{and} \chi d}{{x}^{2} {y}^{2} + x {y}^{3}}$