Complex Fractions

Question

I have been studying Complex Fractions and have made some progress, though am still facing some road blocks. Particularly with finding the LCD.

1/x^2 - 1/y^2 y^2 - x^2 / x^2 y^2

--------------------- I am then told to simplify leading to: -------------------------------

1/x^3 - 1/y^3 y^3 + x^3 / x^3 y^3

I am not exactly sure how they arrived at the numerator. I have been trying to work with the concept of finding the LCD first. Is this the approach i need to take to answer this problem, or at least get to the point that I notated?

Answer 1

1 / x² - 1 / y² LCD = x²y²

x²y² / x²(x²y²) - x²y² / y²(x²y²) multiply numerator and denominator by LCD

y² / (x²y²) - x² / (x²y²) Reduce

(y² - x²) / x²y² Combine

Follow the same process for the denominator.

Copying everything as vertical fractions would clarify the process.

Answer 2

[(1/x^2)-(1/y^2)]/[(1/x^3)-(1/y^3)]

the common denominator of the numerator is x^2y^2

the common denominator of the denominator is x^3y^3

(y^2-x^2)/(x^2y^2)÷(y^3-x^3)/x^3y^3)

dividing by a number is the same as multiplying by its reciprocal, so apply the rule

(y^2-x^2)/(x^2y^2)*(x^3y^3)/(y^3-x^3)

factor

(y-x)(y+x)/(x^2y^2)*(x^3y^3)/(y-x)(y^2+xy+x^2)

now simplify

first cancel the (y-x) to get...

(y+x)/(x^2y^2)*(x^3y^3)/(y^2+xy+x^2)

now cancel with x^3y^3 and x^2y^2 to get xy

now you have (y+x)*(xy)/(y^2+xy+x^2)

(xy)(y+x)/(y^2+xy+x^2) is the simplified answer