Negative Exponents
Negative Exponents with Variables

In Math, ‘integer exponents’ are exponents that are whole numbers, they can also be referred to as ‘integral exponents’.

Before looking at negative exponents and negative exponents with variables examples, we can recap some of the properties of positive integer exponents from the exponents in Algebra page.

Positive Integer Exponents:

The following applies where   x > 0, \space y > 0,   with  a  and  b  being integers.


–   (xy)^a \space = \space x^ay^a

–   (x^{a})^b \space = \space x^{ab}

–   x^{a}x^{b} \space = \space x^{a+b}

–   {\frac{x^a}{x^b}}  =  x^{a{\text{--}}b}

–   ({\frac{x}{y}})^a  =  {\frac{x^a}{y^a}}

–   x^0 \space = \space 1       Provided  x {\cancel{=}} \space 0.

Negative Exponent:

When we encounter a negative exponent in Math, provided  a  is a number not equal to zero, and  m  is a positive integer.

Then:    a^{{\text{-}}m}  =  {\frac{1}{a^m}}         ( a ≠ 0 )


Here,  a  can’t be equal to zero, as you can’t have zero as the denominator on the bottom of a fraction.

Also while we say  m  is a positive integer, there’s no need for any confusion as the minus sign in front of  m  makes it a negative exponent.

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Negative Exponents Examples

(1.1) 

a)    2^{{\text{-}}3}  =  {\frac{1}{2^3}}  =  {\frac{1}{8}}         b)    ({\text{-}}2)^{{\text{-}}3}  =  {\frac{1}{({\text{-}}2)^3}}  =  {\frac{1}{{\text{-}}8}}

c)    t^{{\text{-}}5}  =  {\frac{1}{t^5}}                     d)    r^{{\text{-}}1}  =  {\frac{1}{r}}



(1.2) 

({\frac{x}{y}})^–n  =  ({\frac{y}{x}})ⁿ

When we have a situation like this with a variable fraction in brackets affected by a negative exponent. We flip the fraction and change the exponent to positive, we can show why with numbers.

({\frac{2}{3}})^-1   =   {\frac{1}{\frac{2}{3}}}   =   {\frac{3}{2}}



(1.3) 

{\frac{x^{{\text{-}}a}}{y^{{\text{-}}b}}}  =  {\frac{y^b}{x^a}}

We can show this working with numbers like in example (1.2).

{\frac{3^{{\text{-}}2}}{4^{{\text{-}}3}}}   =   {\frac{\frac{1}{3^2}}{\frac{1}{4^3}}}   =   {\frac{1}{3^2}} ÷ {\frac{1}{4^3}}   =   {\frac{1}{3^2}} × {\frac{4^3}{1}}   =   {\frac{4^3}{3^2}}



(1.4) 

Care needs to be taken when variables are combined, we can look at two typical cases.

a)    xy^{{\text{-}}2}

One may think that this could become  {\frac{1}{xy^2}},  but that’s not quite the case.

The term   xy^{{\text{-}}2}   is the equivalent of   x × {\frac{1}{y^2}},   only the y variable is affected.
So it turns out that   xy^{{\text{-}}2}  =  {\frac{x}{y^2}}.


b)    (xy)^{{\text{-}}2}

This is the term that does become  {\frac{1}{xy^2}},

as the exponent affects everything inside the brackets.

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Simplifying and Negative Exponents

(2.1) 

a)    (a^{{\text{-}}3})^2 \space = \space a^{{\text{-}}2 \times 3} \space = \space a^{{\text{-}}6} \space = \space {\frac{1}{6}}

b)    (x^{{\text{-}}3}y^2)^4 \space = \space a^{{\text{-}}2 \times 3} \space = \space x^{{\text{-}}3 \times 4}y^{2 \times 4} \space = \space x^{{\text{-}}12}y^8



(2.2) 

({\frac{c}{d}})^-2   =   ({\frac{d}{c}})²



(2.3) 

( 4r^{{\text{-}}3})^{{\text{-}}1}

( 4r^{{\text{-}}3})^{{\text{-}}1}   =   {\frac{1}{4r^{{\text{-}}3}}}   =   4r^3

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