Download the exponent laws

MPM2D
WORKING WITH EXPONENTS
Exponents are a simpler way of showing a repeated multiplication. Powers are
products of equal factors. They can be written in expanded form or exponential
form.
3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 can also be written as 38
the entire expression is called a POWER
38
the number 8 is the EXPONENT,
it tells us the number of times
the base has been repeated
the number 3 is the BASE,
it tells us the number that is
being multiplied by itself
THE EXPONENT LAWS
Example 1: Multiply
(4)3 (4)2
if we expand
(4)3(4)2
Notice - the bases are the same
it means
(4 x 4 x 4) (4 x 4)
= 45
Example 2: Multiply
(x)4 (x)2
if we expand
= (x)(x)(x)(x) (x)(x)
Example 3: Multiply
(x2y3)(x3y4)
if we expand
= (x)(x)(y)(y)(y)
= (x)6
(x)(x)(x)(y)(y)(y)(y)
= (x)5(y)7
LAW #1: MULTIPLICATION OF POWERS
To multiply powers with the same base, add the exponents
and keep the base the same
(x)m (x)n = x m+n
Example 4: Divide (4)5 ÷ (4)2
Example 5: Divide (x4) ÷ (x)2
(4)( 4)( 4)( 4)( 4)
(4)( 4)
if we expand
=
reduce
= (4)3
=
( x)( x)( x)( x)
( x)( x)
= (x)2
Example 6: Divide
(x3y4) ÷ (x2y2)
if we expand
=
reduce
= xy2
( x)( x)( x)( y )( y )( y )( y )
( x)( x)( y )( y )
LAW #2: DIVISION OF POWERS
To divide powers, subtract the exponents and
keep the base
(x)m ÷ (x)n = x m-n
Example 7: Expand
(53)2
If we expand
= (53)(53)
= (x2y3)(x2y3)(x2y3)
simplify
= 56
= x6y9
Example 8: Expand
(x2y3)3
3
Example 9: Expand
If we expand
simplify
 x2 
 2 
y 
 x 2  x 2
=  2  2
 y  y
 x6 
=  6 
y 
 x 2 
 2 
 y 
POWER OF A QUOTIENT LAW
POWER OF A POWER LAW
To apply a power to a power, multiply the
exponents and keep the base the same
(xm)n = xmn
To apply a power to a quotient, multiply
the exponents in both the numerator and
the denominator, keep the base the same.
 xm
 m
y
n

 x mn 
   mn 

y 
ZERO AND NEGATIVE EXPONENTS
We can use the patterns and the basic laws of exponents to understand the
meaning of zero and negative exponents.
43 ÷ 4 3
Example 1: Simplify
43 ÷ 43
4 4 4
=
4 4 4
=1
43 ÷ 43
but also
= 4 3-3
= 40
therefore, 40 = 1
Example 2: Simplify
42 ÷ 45
42 ÷ 45
42 ÷ 45
but also
4 4
4 4 4 4 4
1
=
4 4 4
1
= 3
4
= 4 2-5
=
therefore, 4-3 =
= 4 –3
1
43
NEGATIVE EXPONENTS
Any power with a negative exponent can be
expressed in an equivalent fraction form with a
positive exponent
1
n
x5-B-2
 n
x
**Exponents change their sign if they move from numerator to denominator OR
from denominator to numerator! Always try to express answers with positive
exponents only!**
Eg.
1
= 43
4 3
and
 x 2 y 3 
 5 6  =
x y 
EXPONENT WORKSHEET
Use the exponent laws we’ve learned to find the following:
1) State the following numbers as powers of 2
a) 16 _______
b) 2
d) (28)(27) _________
_______
c) 32
e) 2 x 29 ________
f)
_________
28
_________
24
g) 4 x 32 _________ (hint: change to same base) h) 4 x 8 _________
2) State the following numbers as powers of 3
a) 27
d)
34
_______
35
x
_________
b) 243 _________
c) 38 x 3 __________
35
e) 2
3
32
f) 5
3
________
__________
g) 27 x 81 _________
3) Simplify the following expressions
a) (3x )
3 3
d)
f)
_________
710  712  7 6
7 
3 9
2 n 2  4 n1
8n

b)  3x
6
4x 
____________
3
 x2
c)  4
y
__________
( x m  n )( x 2 m )
e)
xn
3

 ________

_____________
____________ (hint: change to the same base)
4) Simplify the following expressions
2 x y 
8 x y x y 
2
a)
4
3
2 5
2
3
__________
b)
2 xy 4 15 x 2 y 3

3x 2 y 2 12 x 4 y 2
_____________
c)
2 n  4 n 1  8 3n  2
16 2 n 1
(HINT: express each with a base of 2)
5) State the following using only positive exponents
a) x 8
________
b) x 2 y 2
1
2 1
________
e)
g) 3 2
________
h) (-3 x 2 y 2 )( 4 x 2 y 2 ) _________
d)
1
3 1
_________
_________
c) a 3 b 4
________
1
2 3
________
f)
6) Evaluate the following
a) 50
_______
1
d) 2  3  4
f)
0
3 8
(express with same base)
h) 5-1 + 5-2
(careful!)
c) (3-2)-2 ________
2 
3 2
2
27 2
g) 2-1 + 3-1
b) 4-2 _________
_________
e)
2 5
__________