simplify:
(x^-2 - y^-2) / (x^-1 - y^-1)
i know you have to get rid of the negative powers, but i cant figure out how, any help?
The numerator can be written
(x^-1 + y^-1)(x^-1 -y^-1)
One of the factors cancels the denominator, leaving you with
(x^-1 + y^-1)
Using a common denominator xy, that can be rewritten
(y + x)/xy
multiply top and bottom by (x^2y^2/(x^2y^2))
so
(x^2y^2/(x^2y^2))[(x^-2 - y^-2)/(x^-1 - y^-1)]
= (y^2 - x^2)/(xy^2 - x^2y)
= (y+x)(y-x)/[xy(y-x)]
= (y+x)/(xy)
I usually check a question like that by letting x and y be some numbers, and subbing that back in the original line and the final line.
Here I tried x=3 and y=4, and it worked.
Wow, drwls saw an easier way.
Nice one!
thank you!
To simplify the given expression, (x^-2 - y^-2) / (x^-1 - y^-1), you need to rationalize the denominators, which means eliminating the negative exponents.
Let's start by simplifying the expression step by step:
Step 1: Rewrite the expression with positive exponents.
(x^-2 - y^-2) / (x^-1 - y^-1) can be rewritten as:
(1/x^2 - 1/y^2) / (1/x - 1/y)
Step 2: Find the common denominator for the fractions on the top and bottom.
The common denominator for (1/x^2 - 1/y^2) is x^2y^2.
Step 3: Rewrite the expression with the common denominator.
((y^2 - x^2)/(x^2y^2)) / ((y - x)/(xy))
Step 4: To divide fractions, invert the second fraction and multiply.
((y^2 - x^2)/(x^2y^2)) * ((xy)/(y - x))
Step 5: Simplify the expression.
((y + x)(y - x))/(xy(y - x))
Step 6: Cancel out the (y - x) terms.
(y + x)/(xy)
Therefore, the simplified expression is (y + x)/(xy).