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Simplifying Expressions with Exponents:
     Complicated Examples
(page 3 of 3)

  • Simplify the following expression:
    • (3^(-1) a^4 b^(-3))^(-2) / (6 a^2 b^(-1) c^(-2))^2

    Before I can cancel anything off, I need to simplify that top parentheses, because it has a negative exponent on it. I can't cancel off, say, the a's, because that a4 isn't really on top. I can either move the whole parentheses down, square, and then simplify, or I can take the negative-square through first. I'll show both ways:

    moving the parentheses first


    squaring first


There are other ways to go about simplifying the above. As long as each step is correct and you get the right answer, your method will be right. Copyright Elizabeth Stapel 2004-2011 All Rights Reserved

  • Simplify the following expression:
    • (3/x)^(-2)

    This is a special case. The negative exponent says that whatever is on top should go underneath, and whatever is underneath should go on top. So I'll just flip the fraction (remembering to change the power from a negative to a positive), and simplify from there:

      (3/x)^(-2) = (x/3)^2 = x^2 / 9

Warning: This only works if the negative exponent is on the whole fraction.

  • Simplify the following expression:
    • [ (15 m^3 n^(-2) p^(-1)) / (25 m^(-2) n^(-4)) ]^(-3)

    There are so many ways I can do this. I'll show four:

    flip inside, simplify, negative cube, flip, and simplify:


    flip inside, simplify, flip the fraction, and cube:


    flip the fraction, simplify inside, cube, flip the negative exponents, and simplify:


    flip the fraction, flip the negative exponents, simplify, and cube:


You should expect to see at least one exercise on your test that is as complicated as this last example. Just take your time, work slowly and carefully, and don't try to do too much at once. If you work bit by bit, you should be able reliably to get the right answer.

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Cite this article as:

Stapel, Elizabeth. "Simplifying Expressions with Exponents: Complicated Examples."
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