Question

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

Answer 1

See explanation

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)`