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Exponents in Algebra

On this Page:
 1. Exponents with Numbers
 2. Exponent Laws
 3. Summary

Exponents in Algebra appear quite often in expressions and equations, as such knowing how to effectively work with them is something that is important when learning Algebra.



Exponents with Numbers:

We can recall how standard exponents work with numbers.
Where the result is a multiplication of the number by itself a certain number of times.

52  =  5 × 5     ,     53  =  5 × 5 × 5

General case:
5n  =  5 × 5 × 5 ……… × 5     n times.     For n > 0.


There are some laws for exponents in Algebra that can be handy to know, sometimes referred to as laws of indices.

These laws for exponents can be particularly useful when manipulating expressions in Algebra that contain one or more variables.

Below they are listed and demonstrated with an example involving numbers.





Laws for Exponents in Algebra


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


1)

(xy)^a \space = \space x^ay^a

EXAMPLE

( 2 × 4 )2  =  82  =  64     ,     22 × 42  =  4 × 16  =  64



2)

(x^{a})^b \space = \space x^{ab}

EXAMPLE

( 22 )3  =  43  =  64     ,     22×3  =  26  =  64



3)

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

EXAMPLE

32 × 33  =  9 × 27  =  243     ,     32+3  =  35  =  243



4)

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

EXAMPLE

{\frac{4^5}{4^3}}  =  {\frac{4 \times 4 \times 4 \times 4 \times 4}{4 \times 4 \times 4}}

Now some 4’s can be cancelled out.

{\frac{{\cancel{4}} \times {\cancel{4}} \times {\cancel{4}} \times 4 \times 4}{\cancel{4} \times \cancel{4} \times \cancel{4}}}   =   4 \times 4  =  4^2  =  16


4^{5{\text{--}}3}  =  4^2  =  16




5)

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

EXAMPLE

({\frac{8}{4}})^2  =  2^2  =  4

{\frac{8^2}{4^2}}  =  {\frac{64}{16}}  =  4



6)

x^0 \space = \space 1

Unless  x \space = \space 0,  then  x^0  is undefined.






Laws for Exponents in Algebra Summary



1)     (xy)^a \space = \space x^ay^a

2)     (x^{a})^b \space = \space x^{ab}

3)     x^{a}x^{b} \space = \space x^{a+b}

4)     {\frac{x^a}{x^b}}  =  x^{a{\text{--}}b}

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

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





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