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

Question

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

Avery Alexander · Accepted Answer

See explanation

Rules used: Negative exponents #1/x^a=x^-a# Exponent of exponent value #(a^b)^c=a^(bxxc)# Exponential multiplication #a^b*a^c=a^(b+c)# Distributive property #a(b+c)=a(b)+a(c)# Zero exponent #z^0=1# Steps: 1. Simplify the first half. Negative exponents #(y-x)/(x^2y)=(y-x)*(x^2y)^-1# Exponent of exponent value #y(x^-2y^-1)-x(x^-2y^-1)# Distributive property #(y-x)(x^-2y^-1)=(x^2y)^-1=x^(2xx(-1))y^-1=x^-2y^-1# Exponential multiplication #y^(1-1)x^-2-x^(1-2)y^-1=y^0x^-2-x^-1y^-1# Zero exponent #(1)x^-2-x^-1y^-1# Negative exponent #1/x^2-1/(xy)# Result: #\color(tomato)(1/x^2-1/(xy))# 2. Simplify the second half. Distributive property #(x+y)(xy^2)=x(xy^2)+y(xy^2)# Exponential multiplication #x(xy^2)=x^(1+1)y^2=x^2y^2# and #y(xy^2)=xy^(1+2)=xy^3# Result: #\color(orchid)(x^2y^2+xy^3)# 3. Combine these simplified halves. #\color(tomato)(1/x^2-1/(xy))+\color(orchid)(x^2y^2+xy^3)#

Jameson Allen · Answer

undefined To simplify the expression (y – x)/(x^2y) + (x + y)/(xy^2), we can find a common denominator and combine the fractions. The common denominator is x^2y * xy^2 = x^3y^3.

Multiplying the first fraction by y^2/y^2 and the second fraction by x^3/x^3, we get (y^3 - xy^2)/(x^3y^3) + (x^3 + xy^3)/(x^3y^3).

Combining the numerators, we have (y^3 - xy^2 + x^3 + xy^3)/(x^3y^3).

Simplifying further, we get (x^3 + y^3 + xy^3 - xy^2)/(x^3y^3).

This is the simplified form of the expression.