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

3 Answers

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  • Pat
    Lv 4
    9 years ago
    Favorite Answer

    (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

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