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Negative Exponents (page 2 of 5)

Sections: Basics, Negative exponents, Scientific notation, Engineering notation, Fractional exponents

A negative exponent just means that the base is on the wrong side of the fraction line, so you need to flip the base to the other side. For instance, "x–2" just means "x2, but underneath, as in 1/(x2)".

  • Write x–4 using only positive exponents.

      • x^(-4) = 1/x^4

  • Write x2 / x–3 using only positive exponents.

      • (x^2) / (x^(-3)) = (x^2)(x^3) = x^5

  • Write 2x–1 using only positive exponents.

      • 2x^(-1) = 2/x^1 = 2/x 

Note that the "2" above does not move with the variable; the exponent is only on the "x".

  • Write (3x)–2 using only positive exponents.

      • ( 3x )^(-2) = 1/( 3x )^2 = 1/(9x^2)

  • Write (x–2 / y–3)–2 using only positive exponents.

      • [ x^(-2) / y^(-3) ]^(-2) = [ y^(-3) / x^(-2) ]^2 = [ y^(-6) ] / [ x^(-4) ] = (x^4)/(y^6)

    This one can also be done as:   Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

      [ x^(×2) / y^(-3) ]^(-2) = [ x^(-2) ]^-2 / [ y^(-3) ]^(-2) = (x^4) / (y^6)

Order is kinda flexible with this stuff...

By the way, now that you know about negative exponents, you can understand the logic behind the "anything to the power zero" rule:

    Anything to the power zero is just "1".

Why is this so? There are various explanations. One might be stated as "because that's how the rules work out." Another would be to trace through a progression like the following:

    35 = 36 ÷ 3 = 243
    3    4 = 35 ÷ 3 = 81
    3    3 = 34 ÷ 3 = 27
    3    2 = 33 ÷ 3 = 9
    3    1 = 32 ÷ 3 = 3

Then logically 30 = 31 ÷ 3 = 3 ÷ 3 = 1.

A negative-exponents explanation might be as follows:

    m0 = m(n – n) = mn × m–n = mn ÷ mn = 1

...since anything divided by itself is just "1".

Another comment: Please don't ask me to "define" 00. There are at least two ways of looking at this quantity:

  • Anything to the zero power is "1", so 00 = 1.
  • Zero to any power is zero, so 00 = 0.

As far as I know, the "math gods" have not yet settled on a "definition" of 00. In fact, in calculus, "00" will be called an "indeterminant form". If this quantity comes up on class, don't assume: ask your instructor what you should do with it.

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Cite this article as:

Stapel, Elizabeth. "Negative Exponents." Purplemath. Available from
    http://www.purplemath.com/modules/exponent2.htm. Accessed
 

 

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