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Simplifying Expressions with Exponents (page 1 of 3)

To simplify with exponents, don't feel like you have to work only from the rules for exponents. It is often simpler to work directly from the definition and meaning of exponents. For instance:

  • Simplify x6 × x5
  • The rules tell me to add the exponents. But I when I started algebra, I had trouble keeping the rules straight, so I just thought about what exponents mean. The " x6 " means "six copies of x multiplied together", and the " x5 " means "five copies of x multiplied together". So if I multiply those two expressions together, I will get eleven copies of x multiplied together. That is:

      x6 × x5 = (x6)(x5)  
                  = (xxxxxx)(xxxxx)    
      (6 times, and then 5 times) 
                  = xxxxxxxxxxx         
      (11 times)  
                  = x11


      x6 × x5 =   x11

  • Simplify the following expression:
    • 6^8 / 6^5

    The exponent rules tell me to subtract the exponents. But let's suppose that I've forgotten the rules again. The " 68 " means I have eight copies of 6 on top; the " 65 " means I have five copies of 6 underneath.

      (6×6×6×6×6×6×6×6) / (6×6×6×6×6)

    How many extra 6's do I have, and where are they? I have three extra 6's, and they're on top. Then:

      (6×6×6) / 1 = 6^3

  • Simplify the following expression:
    • t^10 / t^8

    How many extra copies of t do I have, and where are they? I have two extra copies, on top:

      (t×t)/1 = t^2

  • Simplify the following expression: Copyright © Elizabeth Stapel 2004-2011 All Rights Reserved
    • 5^3 / 5^9

    How many extra copies of 5 do I have, and where are they? I have six extra copies, underneath:

      1/(5×5×5×5×5×5) = 1 / 5^6

Note: If you apply the subtraction rule, you'll end up with 53–9 = 5–6, which is mathematically correct, but is almost certainly not the answer they're looking for. Whether or not you've been taught about negative exponents, when they say "simplify", they mean "simplify the expression so it doesn't have any negative or zero powers". Some students will try to get around this minus-sign problem by arbitrarily switching the sign to magically get " 56 " on top (rather than below a "1"), but this is incorrect.

  • Simplify the following expression:
    • (5x^5) / (3x^3)

    Don't forget that the "5" and the "3" are just numbers. Since 3 doesn't go evenly into 5, I can't cancel the numbers. Don't try to subtract the numbers, because the 5 and the 3 in the
    " 5/3 " are not at all the same as the 5 and the 3 in rational expression " x5 / x3 ". The
    5/3 stays as it is.

    For the variables, I have two extra copies of x on top, so the answer is:

      (5xx) / 3 = (5x^2) / 3 = (5/3)x^2

Either of the purple highlighted answers should be acceptable: the only difference is in the formatting; they mean the same thing.




  • Simplify (–46x2y3z)0
  • This is simple enough: anything to the zero power is just 1.

      (–46x2y3z)0 = 1

  • Simplify –(46x2y3z)0
  • The parentheses still simplifies to 1, but this time the "minus" is out front, out from under the power, so the exponent doesn't touch it. So the answer is:

      –(46x2y3z)0 = –1

  • Simplify the following expression:
    • (15a^5 b^2 c^4) / (25a^3 b^3 c^4)

    I can cancel off the common factor of 5 in the number part of the fraction:

      (3a^5 b^2 c^4) / (5a^3 b^3 c^4)

    Now I need to look at each of the variables. How many extra of each do I have, and where are they? I have two extra a's on top. I have one extra b underneath. And I have the same number of c's top and bottom, so they cancel off entirely. This gives me:


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Cite this article as:

Stapel, Elizabeth. "Simplifying Expressions with Exponents." Purplemath. Available from Accessed



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