Exponents: Negative exponents

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Introduction
Lessons
  1. What does a Negative Exponent do?
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Examples
Lessons
  1. Analyzing Numbers With Negative Exponents
    Evaluate:
    i)
    424^{-2}
    ii)
    161216^{-\frac{1}{2}}
    iii)
    642364^{-\frac{2}{3}}
    iv)
    8134-81^{-\frac{3}{4}}
    v)
    (27)23(-27)^{-\frac{2}{3}}
    1. Analyzing Expressions With Negative Exponents
      Simplify:
      i)
      (3x2y3)2x1y2\frac{(-3x^2y^{-3})^{-2}}{x^{-1}y^2}
      ii)
      1253x12512x(15)2x3\frac{125^{3x-1}\cdot25^{1-2x}}{(\frac{1}{5})^{2x-3}}
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      Topic Notes

      What is negative exponent?

      A negative exponent helps to show that a base is on the denominator side of the fraction line.  In other words, the negative exponent rule tells us that a number with a negative exponent should be put to the denominator, and vice versa. For example, when you see x^-3, it actually stands for 1/x^3. Not too bad right?

       

      You might be wondering about the fraction line, since there isn't one when we just look at x^-3. However, you can actually convert any expression into a fraction by putting 1 over the number. That's the main reason why we can move the exponents around and solve the questions that are to follow.

       

      Learning this lesson will also help you get one step closer to understanding why any number with a 0 in its exponent equals to 1. There'll be a link to a chart at the end of this lesson that can show you how that relationship comes about. You'll soon understand all the basic properties of exponents!

       

      How to solve for for negative exponents

      Let's try working with some negative exponent questions to see how we'll move numbers to the top or bottom of a fraction line in order to make the negative exponents positive. We'll start with regular numbers with a negative exponent, then move on to fractions that have negative exponents on both its numerator and denominator.

       

      Question 1:

      Solve 2^-2

       

      Solution:

      As we learned earlier, if we move the number to the denominator, it'll get rid of the negative in the exponent. Then, solving for exponents is easy once we have it in a more calculation-friendly form.

      1/(2^2)

      = 1/4

       

      Question 2:

      Solve -2^-2

       

      Solution:

      In this case, we've got a negative number with a negative exponent. Again, just move the number to the denominator of a fraction to make the exponent positive. One way you can rewrite the question we're given is the following:

      -2^-2

      = (-1)(2^-2)

       

      Multiplying in that -1 will turn the equation back into what it was originally. However, keeping the -1 outside helps us work with the negative exponent a little easier and allows us to illustrate what's happening.

       

      So moving on from the above, we can continue solving with the negative exponent as we did before.

      (-1)(2^-2)

      = (-1)(1/2^2)

      = (-1)(1/4)

      = -1/4

       

      As you can see, the final answer we get is negative!.

       

      Fractions with negative exponents

       

      Question 3:

      (3^-2)/(4^-3)

       

      Solution:

      If you ever see a negative exponent on the top of a fraction, you know that if you flip it to the bottom, it'll become positive. The same actually works for negative exponents on the bottom. If you move it to the numerator, its exponent also becomes positive. With that in mind, let's work through the question. Our first step is just to flip the numerator and denominator to get rid of all the negatives in the exponents. Then solve as usual with the power rule.  

       

      (3^-2)/(4^-3)

      = (4^3)/(3^2)

      =64/9

       

      Definitely not as confusing as it first looked, right?  

       

      Here's a good place to take a look at comparing negative and positive exponents and seeing how they behave on a graph.

      a1=1aa^{-1}=\frac{1}{a}
      amn=1amna^{-\frac{m}{n}}=\frac{1}{a^{\frac{m}{n}}}