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Laws of Exponents

Let us recall the laws of exponents.

Law 1: Multiplying Powers with the Same Base

m
n
If a be a non-zero integer and m, n are two whole numbers, then a × a = a m + n .
7
5
For example, 3 × 3 = 3 7 + 5 = 3 12
Law II: Dividing Powers with the Same Base

If a be a non-zero integer and m, n are two whole
Think and Answer
a m
n
m
numbers, then a ÷ , a = = a mn where m > n.
−
a n If the equation for bacteria growth
t
is given by N = 2 , where N is the count
7 7
−
7
For example, 7 ÷ 7 3 = = 7 73 = 7 4 of bacteria after t hours, then answer the
7 3 following questions:
Law III: Taking Power of a Power 1. After how many hours the number of
bacteria is 32?
If a is any non-zero integer and m, n are whole 2. What will be the number of bacteria after
m n
mn
numbers, then (a ) = a . 10 hours?
For example, (–5 ) = (–5) 3 × 4 = (–5) .
3 4
12
Law IV: Multiplying Powers with the Same Exponents

m
m
m
If a and b are two non-zero integers, then a × b = (ab) , where m is any whole number.
4
4
4
4
For example, (–3) × 2 = {(–3) × 2} = (–6) .
Law V: Dividing Powers with the Same Exponents Get it right!
a m  a m 1.  x a ×  x b =  x a × b
m
m
For any two non-zero integers a and b, a ÷ b = =   ,      
y

y
y


b m  b
where m is any whole number.    a ×    b =    a + b
x
x
x
y

y

y

7
7
For example, − ( ) ÷− ( ) = − ( ) 4 7 =  −  4  7 =   4 7 2.  x a  x b  x a ÷ b
4
3
 

− ( ) 3 7  −  3   3   ÷   =  

y
y

y

–
x
 2 3  2 4  2 2 + 1  x a  x b  x a b
Example 1: Find the value of x if   ×   =     ÷   =  

y
y

y

7 

7 
7 


x
2
Solution: We have  2 3 ×  2 4 =    2 + 1
 
 

7 
7 

7 

x
x
 2 3 + 4  2 2 + 1  2 7  2 2 + 1
or   =   ⇒   =  

7 

7 


7 
7 
⇒ 2x = 7 – 1 = 6 ⇒ x = 3
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