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negative exponents and negative numbers

If you have -3 with a -4 exponent

 

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Erica H. | Patient and Knowledgeable Tutoring By Erica H.Patient and Knowledgeable Tutoring By Er...
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Tamara J. | Math Tutoring - Algebra and Calculus (all levels)Math Tutoring - Algebra and Calculus (al...
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Negative exponents have the following property:   n-m = 1/nm   OR   1/n-m = nm

So for  (-3)-4 ,  n= -3 and -m= -4 (or m= 4).

Thus,  (-3)-4 = 1/(-3)4

Jordan K. | Nationally Certified Math Teacher (grades 6 through 12)Nationally Certified Math Teacher (grade...
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We can rewrite (-3-4) using a positive exponent by taking the reciprocal of the base (-3) and applying the absolute value of the negative exponent (-4) to the denominator of the fraction like this:  1/(-3)4.  Now we can simplify the denominator by evaluating (-3)4 which is the same as writing (-3)(-3)(-3)(-3) giving us 81.  So our fraction in simplest form is 1/81.  One interesting thing to note here is that raising a negative number by an even numbered exponent gives a positive value (as in our case) and raising a negative number by an odd numbered exponent yields a negative result.