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Laws of Exponents

Module by: Kenny M. Felder. E-mail the author

Summary: This module covers the laws the govern the manipulation of exponents in algebra.

The following are generally referred to as the “laws” or “rules” of exponents.

x a x b = x a + b x a x b = x a + b size 12{x rSup { size 8{a} } x rSup { size 8{b} } =x rSup { size 8{a+b} } } {}
(1)
x a x b = x a b or 1 x b a x a x b = x a b or 1 x b a size 12{ { {x rSup { size 8{a} } } over {x rSup { size 8{b} } } } =x rSup { size 8{a - b} } " or " { {1} over {x rSup { size 8{b - a} } } } } {}
(2)
( x a ) b = x ab ( x a ) b = x ab size 12{ \( x rSup { size 8{a} } \) rSup { size 8{b} } =x rSup { size 8{ ital "ab"} } } {}
(3)

As with any formula, the most important thing is to be able to use them—that is, to understand what they mean. But it is also important to know where these formulae come from. And finally, in this case, the three should be memorized.

So...what do they mean? They are, of course, algebraic generalizations—statements that are true for any xx size 12{x} {}, aa size 12{a} {}, and bb size 12{b} {} values. For instance, the first rule tells us that:

7 12 7 4 = 7 16 7 12 7 4 = 7 16 size 12{7 rSup { size 8{"12"} } cdot 7 rSup { size 8{4} } =7 rSup { size 8{"16"} } } {}
(4)

which you can confirm on your calculator. Similarly, the third rule promises us that

( 7 12 ) 4 = 7 48 ( 7 12 ) 4 = 7 48 size 12{ \( 7 rSup { size 8{"12"} } \) rSup { size 8{4} } =7 rSup { size 8{"48"} } } {}
(5)

These rules can be used to combine and simplify expressions.

Example 1

Table 1
Simplifying with the Rules of Exponents
x 3 4 x 5 x 9 x 11 x 3 4 x 5 x 9 x 11 size 12{ { { left (x rSup { size 8{3} } right ) rSup { size 8{4} } cdot x rSup { size 8{5} } } over {x rSup { size 8{9} } cdot x rSup { size 8{"11"} } } } } {}  
= x 12 x 5 x 9 x 11 = x 12 x 5 x 9 x 11 size 12{ {}= { {x rSup { size 8{"12"} } cdot x rSup { size 8{5} } } over {x rSup { size 8{9} } cdot x rSup { size 8{"11"} } } } } {} Third rule: (xa)b=xab(xa)b=xab size 12{ \( x rSup { size 8{a} } \) rSup { size 8{b} } =x rSup { size 8{ ital "ab"} } } {}, wherea=3a=3 size 12{a=3} {}andb=4b=4 size 12{b=4} {}
= x 17 x 20 = x 17 x 20 size 12{ {}= { {x rSup { size 8{"17"} } } over {x rSup { size 8{"20"} } } } } {} First rule: xaxb=xa+bxaxb=xa+b size 12{x rSup { size 8{a} } x rSup { size 8{b} } =x rSup { size 8{a+b} } } {}, done on both the top and bottom
= 1 x 3 = 1 x 3 size 12{ {}= { {1} over {x rSup { size 8{3} } } } } {} Second rule: xaxb=1xbaxaxb=1xba size 12{ { {x rSup { size 8{a} } } over {x rSup { size 8{b} } } } = { {1} over {x rSup { size 8{b - a} } } } } {}, where we choose this form to avoid a negative exponent

Why do these rules work? It’s very easy to see, based on what an exponent is.

Table 2
Why does the first rule work?
19 3 19 3 size 12{"19" rSup { size 8{3} } } {} 19 4 19 4 size 12{"19" rSup { size 8{ cdot 4} } } {}
= ( 19 19 19 ) = ( 19 19 19 ) size 12{ {}= \( "19" cdot "19" cdot "19" \) } {} ( 19 19 19 19 ) ( 19 19 19 19 ) size 12{ cdot \( "19" cdot "19" cdot "19" \) } {}
= 19 7 = 19 7 size 12{ {}="19" rSup { size 8{7} } } {}  

You see what happened? 193193 size 12{"19" rSup { size 8{3} } } {} means three 19s multiplied; 194194 size 12{"19" rSup { size 8{ cdot 4} } } {} means four 19s multiplied. Multiply them together, and you get seven 19s multiplied.

Table 3
Why does the second rule work?
First form Second form
19 8 19 5 19 8 19 5 size 12{ { {"19" rSup { size 8{8} } } over {"19" rSup { size 8{5} } } } } {} 19 5 19 8 19 5 19 8 size 12{ { {"19" rSup { size 8{5} } } over {"19" rSup { size 8{8} } } } } {}
= 19 19 19 19 19 19 19 19 19 19 19 19 19 = 19 19 19 19 19 19 19 19 19 19 19 19 19 size 12{ {}= { {"19" cdot "19" cdot "19" cdot "19" cdot "19" cdot "19" cdot "19" cdot "19"} over {"19" cdot "19" cdot "19" cdot "19" cdot "19"} } } {} = 19 19 19 19 19 19 19 19 19 19 19 19 19 = 19 19 19 19 19 19 19 19 19 19 19 19 19 size 12{ {}= { {"19" cdot "19" cdot "19" cdot "19" cdot "19"} over {"19" cdot "19" cdot "19" cdot "19" cdot "19" cdot "19" cdot "19" cdot "19"} } } {}
Expanded for of 19 to the eighth over 19 to the fifth, matching 19's are crossed out. Expanded for of 19 to the fifth over 19 to the eighth, matching 19's are crossed out.
= 19 3 = 19 3 size 12{ {}="19" rSup { size 8{3} } } {} = 1 19 3 = 1 19 3 size 12{ {}= { {1} over {"19" rSup { size 8{3} } } } } {}

In this case, the key is fraction cancellations. When the top is multiplied by 19 and the bottom is multiplied by 19, canceling these 19s has the effect of dividing the top and bottom by 19. When you divide the top and bottom of a fraction by the same number, the fraction is unchanged.

You can also think of this rule as the inevitable consequence of the first rule. If 193195=198193195=198 size 12{"19" rSup { size 8{3} } cdot "19" rSup { size 8{5} } ="19" rSup { size 8{8} } } {}, then 198195198195 size 12{ { {"19" rSup { size 8{8} } } over {"19" rSup { size 8{5} } } } } {} (which asks the question “ 195195 size 12{"19" rSup { size 8{5} } } {} times what equals 198198 size 12{"19" rSup { size 8{8} } } {}?”) must be 193193 size 12{"19" rSup { size 8{3} } } {}.

Table 4
Why does the third rule work?
19 3 4 19 3 4 size 12{ left ("19" rSup { size 8{3} } right ) rSup { size 8{4} } } {}
= 19 19 19 19 19 19 19 19 19 19 19 19 = 19 19 19 19 19 19 19 19 19 19 19 19 size 12{ {}= left ("19" cdot "19" cdot "19" right ) cdot left ("19" cdot "19" cdot "19" right ) cdot left ("19" cdot "19" cdot "19" right ) cdot left ("19" cdot "19" cdot "19" right )} {}
= 19 12 = 19 12 size 12{"19" rSup { size 8{"12"} } } {}

What does it mean to raise something to the fourth power? It means to multiply it by itself, four times. In this case, what we are multiplying by itself four times is 193193 size 12{"19" rSup { size 8{3} } } {}, or 191919191919 size 12{ left ("19" cdot "19" cdot "19" right )} {}. Three 19s multiplied four times makes twelve 19s multiplied.

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