Exponents and powers by arjun rastogi

EXPONENTS
• A quantity representing the power to which
a given number or expression is to be
raised, usually expressed as a raised
symbol beside the number or expression
(e.g. 3 in 23 = 2 × 2 × 2).

General Enquiry
• Exponents are shorthand for repeated
multiplication of the same thing by itself.
For instance, the shorthand for multiplying
three copies of the number 5 is shown on
the right-hand side of the "equals" sign
in (5)(5)(5) =53. The "exponent", being 3 in
this example, stands for however many
times the value is being multiplied. The
thing that's being multiplied, being5 in this
example, is called the "base".

Exponents
 35 Power
exponent
base
3 3 means that is the exponential
Example:
form of t
125 5 5
he number
125.

53 means 3 factors of 5 or 5 x 5 x 5

The Laws of Exponent
Comes From 3 ideas
• The exponent says how many times to
use the number in a multiplication.
• A negative exponent means divide,
because the opposite of multiplying is
dividing
• A fractional exponent like 1/n means
to take the nth root:

Laws Of Exponent
• x1 = x
• x0 = 1
• x-1 = 1/x
• xmxn = xm+n
• xm/xn = xm-n
• (xm)n = xmn
• (xy)n = xnyn
• (x/y)n = xn/yn
• x-n = 1/xn

The Laws of Exponents:
#1: Exponential form: The exponent of a power indicates
how many times the base multiplies itself.
n
x  x  x  x  x  x  x 
x
n 
times
n factors of x
Example: 53  555

#2: Multiplying Powers: If you are multiplying Powers
with the same base, KEEP the BASE & ADD the EXPONENTS!
m n m n x x x   
So, I get it!
When you
multiply
Powers, you
add the
exponents!
2 6  2 3  2 6 
3  29
512


#3: Dividing Powers: When dividing Powers with the
same base, KEEP the BASE & SUBTRACT the EXPONENTS!
m
m n m n
n
x
x x x
x
   
So, I get it!
When you
divide
Powers, you
subtract the
exponents!
6
2 6 2 4
  
2 2
16
2
2


#4: Power of a Power: If you are raising a Power to an
exponent, you multiply the exponents!
 n
xm  xmn
So, when I
take a Power
to a power, I
multiply the
exponents
3 2 3 2 5 (5 )  5  5 

#5: Product Law of Exponents: If the product of the
bases is powered by the same exponent, then the result is a
multiplication of individual factors of the product, each powered
by the given exponent.
 n xy  xn  yn
So, when I take
a Power of a
Product, I apply
the exponent to
all factors of
the product.
2 2 2 (ab)  a b

#6: Quotient Law of Exponents: If the quotient of the
bases is powered by the same exponent, then the result is both
numerator and denominator , each powered by the given exponent.
n n
x x
y y
n
 
  
 
So, when I take a
Power of a
Quotient, I apply
the exponent to
all parts of the
quotient.
16
81
4 2
4
  

3
2
3
4




Try these:
  
2 5 1. 3
  
3 4 2. a
  
3. 2a
2 3 4.  2 2 a 5 2 b
3  
5. (  3a 2 )
2  6.  2 3 s t
4  

 




5
7.
s
t

  


 

2
9
3
5
3
8.

  


 

2
8
4
9.
st
rt

  


 

2
5 8
a b
4 5
36
4
10.
a b

SOLUTIONS
  
2 5 1. 3
  
3 4 2. a
  
2 3 3. 2a
  
2 5 3 2 4. 2 a b
  2 2 5. ( 3a )
  
2 4 3 6. s t
10 3
12a
3 2 3 6 2 a  8a 
2 2 5 2 3 2 4 10 6 10 6 2 a b  2 a b 16a b   
  2 2 2 4  3  a  9a 
2 3 4 3 6 12 s t  s t  

SOLUTIONS

 




5
7.
s
t

  


 

2
9
3
5
3
8.

  


 

2
8
4
9.
st
rt

  


 

2
5 8
a b
4 5
36
4
10
a b
2 8
5
5
s
t
 4 2 8 3  3
s t
2
2
4
r
st
r

  


 

 3 2 2 2 3 2 2 6 9ab  9 a b  81a b 

#7: Negative Law of Exponents: If the base is powered
by the negative exponent, then the base becomes reciprocal with the
positive exponent.
1 m
m x
x
  So, when I have a
Negative Exponent, I
switch the base to its
reciprocal with a
Positive Exponent.
Ha Ha!
If the base with the
negative exponent is in
the denominator, it
moves to the
numerator to lose its
negative sign!
1
1
 
3 9

and
1
3
125
5
5
2
2
3
3
 


#8: Zero Law of Exponents: Any base powered by zero
exponent equals one.
0 1 x 
0
5 1
1
and
0
0

(5 ) 
1

a
a
and
So zero
factors of a
base equals 1.
That makes
sense! Every
power has a
coefficient
of 1.

Try these:
  
2 0 1. 2a b
  2 4 2. y y
  
5 1 3. a
4. s 2  4s
7   2 4 5. 3x y
3  
  
2 4 0 6. s t

SOLUTIONS
  
2 0 1. 2a b
  
5 1 3. a
1
a
  2 7 4. s 4s
  
2 3 4 5. 3x y
  
2 4 0 6. s t
1
5
4s
5   x
8
4 8 12
12
81
3
y
x y   
1

Exponents and powers by arjun rastogi

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