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Exponents and Negative Numbers (page 4 of 4)

Sections: Introduction, Adding and subtracting, Multiplying and dividing, Negatives and exponents

Now you can move on to exponents, using the cancelation-of-minus-signs property of multiplication. For instance, (3)2 = (3)(3) = 9. In the same way:

  • Simplify (3)2
    • (3)2 = (3)(3) = (+3)(+3) = 9

Note the difference between the above exercise and the following:

  • Simplify 32
    • 32 = (3)(3) = (1)(9) = 9

In the second exercise, the square (the "to the power 2") was only on the 3; it was not on the minus sign. Those parentheses make all the difference in the world! Be careful with them, especially when you are entering expressions into software. Different software may treat the same expression very differently, as one researcher has demonstrated very thoroughly.

  • Simplify (3)3
    • (3)3    = (3)(3)(3)  
                  = (+3)(+3)(3)
                  = (9)(3)
                  = 27

  • Simplify (3)4
    • (3)4    = (3)(3)(3)(3)  
                  = (+3)(+3)(3)(3)
                  = (+3)(+3)(+3)(+3)
                  = (9)(9)
                  = 81

  • Simplify (3)5
    • (3)5    = (3)(3)(3)(3)(3)  
                  = (+3)(+3)(3)(3)(3)
                  = (+3)(+3)(+3)(+3)(3)
                  = (9)(9)(3)
                  = 243

Note the pattern: A negative number taken to an even power gives a positive result (because the pairs of negatives cancel), and a negative number taken to an odd power gives a negative result (because, after canceling, there will be one minus sign left over). So if they give you an exercise containing something slightly ridiculous like (1)1001, you know that the answer will either be +1 or 1, and, since 1001 is odd, then the answer must be 1.   Copyright Elizabeth Stapel 1999-2011 All Rights Reserved

You can also do negatives with roots, but only if you're careful. You can do sqrt(16), because there is a number that squares to 16. That is,

    sqrt(16) = 4

...because 42 = 16. But what about sqrt(-16)? Can you square anything and have it come up negative? No! So you cannot take the square root (or the fourth root, or the sixth root, or the eighth root, or any other even root) of a negative number. On the other hand, you can do cube roots of negative numbers. For instance:

    cbrt(-8) = -2

...because (2)3 = 8. For the same reason, you can take any odd root (third root, fifth root, seventh root, etc.) of a negative number.

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Cite this article as:

Stapel, Elizabeth. "Exponents and Negative Numbers." Purplemath. Available from Accessed


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