# Need explanation! (r^-1+s^-1)^-1?

simplifying complex fractions

(r^-1+s^-1)^-1

1/y+1/y-1 over 1/y-2/y-1

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

Relevance
• Pat
Lv 4
9 years ago

(1/r + 1/s)^-1

Get a common denominator: [(s+r)/sr]^-1

Take the reciprocal: sr/(s+r)

In the numerator, get a common denominator: 1/y+1/y-y/y = (2-y)/y

In the denominator, get a common denominator: 1/y-2/y-y/y = (-2-y)/y

Since the denominator is a fraction, "flip that guy and multiply":

(2-y)/y * y/(-2-y) = (2-y)/(-2-y)

You cold factor a negative out of the denominator, then distribute it through the numerator to get:

(y-2)/(y+2)

In the numerator you've got: 1/x-1/y. Get a common denominator: (y-x)/xy

In the denominator you've got: 1/x² - 1/y², Get a common denominator: (y²-x²)/x²y²

Again, the denominator is a fraction so flip that guy and multiply:

(y-x)/xy * x²y²/(y²-x²) = xy(y-x)/(y²-x²)

Here, the denominator factors to (y-x)(y+x) and the (y-x) cancels to leave:

xy/(y+x)

Or, for the last one, did you mean 1/x - y^-1/x^-2 - 1/y²?

In that case, you would have 1/x - x²/y - 1/y².

Get a common denominator: (y² - x³y - x)/(xy²)

This doesn't simplify as nicely though, so I assumed you meant what I did above.

• 9 years ago

(r^-1+s^-1)^-1 = 1/(1/r + 1/s) = rs/(s+r)

Can you finish the other two?

• Ari
Lv 6
9 years ago

(r^-1+s^-1)^-1

= .....1

_________

...1.......1

___ + ____

...r.........s

the best way to simplify this is to multiply everything by the LCD, which in this case is r* s

= .....1 * r *s

____________

...1*r*s.......1*r*s

______ + ____

...r.............s

r*s / r = s and s*r/s = r

cancel everything possible

..r*s

____

s + r

2.

...1........1

..__ + ____

...y.......y-1

___________

..1........ 2

___ - _____

..y........y-1

again multiply everything by the LCD, which now is y * (y-1)

...1* y(y-1)........1* y (y-1)

.._______ + ....________

...........y...............y-1

____________________

..1 * y (y-1)........ 2 * y( y-1)

_________ - ____________

........y...................y-1

the y's cancel and the y-1 's cancel:

y - 1 + y

_________

y - 1 - 2y

simplify:

2y - 1

______

- y - 1