Laws of Exponents Learning Strategies
What should students be able to do within this interactive?
Students should be able to understand and use of the laws of exponents.
Students should be able to simplify expressions that have positive, negative, or
zero exponents with variable or numeric bases.
Common mistakes made by students:
not understanding the meaning of an exponent as the number of times a
base is repeated in multiplication
manipulating the base instead of the exponent
not remembering that the laws for exponents are shortcuts performed on
the exponents, not the base
thinking that negative exponents produce negative results
not distinguishing one law from another
not distinguishing the difference between a negative number and a
negative exponent
Common Mistakes Made within Each Law:
xm xn
For the Product of Powers:
multiplying the exponents
multiplying or adding the bases
xm
n
xm xn xm n
For the Quotient of Powers:
dividing the exponents
dividing the bases
subtraction errors with negative exponents
For the Power of a Power:
( x m ) n x mn
multiplying the outer exponent onto both the base and the inner exponent
adding the exponents
multiplication errors with negative exponents.
For the Product of a Power:
( x y) m x m y m
not multiplying the outer exponent onto all of the inner exponents
adding the exponents
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n
x
xn
For the Quotient of a Power:
y
yn
not multiplying the outer exponent onto exponents of both the numerator and
denominator.
multiplication errors
dividing exponents
multiplying the exponent and the base together
For Zero Exponents:
x 0 1, x 0
thinking the result of a zero exponent is zero
thinking the result of a zero exponent is the value of the base
not understanding the meaning of a zero exponent and where it could arise in
a question
not understanding why the result of a zero exponent is always one
1
,x 0
xn
thinking that all negative exponents result in negative answers
not making the numerator 1 when reciprocating the base
not understanding the meaning of a negative exponent
forgetting to make the exponent positive after it has been reciprocated
not distinguishing between a negative base and negative exponent
For Negative Exponents:
x
n
Curriculum Connections:
 Please note all of the following correlations match outcomes in the new
Mathematics Kindergarten to Grade 9 Program of Studies (2007).
Grade 9 Number SO1: Demonstrate an understanding of powers with integral bases
(excluding base 0) and whole number exponents by:
representing repeated multiplication, using powers
using patterns to show that a power with an exponent of zero is equal to one
solving problems involving powers.
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Grade 9 Number SO2: Demonstrate an understanding of operations on powers with
integral bases (excluding base 0) and whole number exponents:
Math 10-C Algebra and Numbers SO3: Demonstrate an understanding of powers with
integral and rational exponents.
Print Activity notes:
*Note: The Print Activity is not intended to be an assessment piece
It is recommended that students use the “Explore It” mode to work through the Print
Activity. Students will be asked to examine each law in its exponential and expanded
form. The activity will allow students to see the patterns that form the basis behind the
rules for the laws.
The Print Activity may be opened in Word Format instead of PDF so that changes to
questions can be made.
Laws of Exponents Print Activity Key
1.
Use the Product Law in the “Explore It” mode for the following
exercise.
Move the slider bars as directed:
m (purple) to 2
n (orange) to 3
x (red) to 4
a.
Use the above example in the “Explore It” mode to complete the
following:
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Exponential Form
Expanded Form
4 2 43
4 2 43
(4)(4) (4)(4)(4)
(4)(4)(4)(4)(4)
( 4) 2 3
( 4) 5
b.
xm
Eg. 31
(4) 5
Complete all the missing parts in the following table:
xn
34
Exponential Form
Expanded Form (simplified)
(3)1+4 = (3)5
(3)(3)(3)(3)(3)
(5) 2 (5) 3
(5)2+3 = (5) 5
(5)(5)(5)(5)(5)
( 4) 2 ( 4) 2
( 4) 2 2
(-4)(-4)(-4)(-4)
Eg. (3)-2
(3)-2+4 = (3)2
(3)4
( 2) 1 ( 2) 3
( 2) 1
( 4) 4
3
(3)(3)
( 2) 4
1
( 2)( 2)( 2)( 2)
( 6) 3 ( 6) 2
(-6)3+2 = ( 6) 5
( 6)( 6)( 6)( 6)( 6)
(2)4 (2)-3
(2) 4
2
c.
3
(2)1
Complete the Product Law:
When multiplying like bases, you must add the exponents.
2.
Use the Quotient Law in the “Explore It” mode for the following exercise.
Move the slider bars as directed:
m (purple) to 5
n (orange) to 3
x (red) to 4
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Answer the following using the above example in the “Explore It” mode:
a.
b.
xm
Exponential Form
Expanded Form
45
( 4) 5 3
45 43
(4)(4)(4)(4)(4) (4)(4)(4)
( 4) 2
(4) 2
43
Complete all the missing parts in the following table:
xn
Exponential Form
Expanded Form (simplified)
(3)(3)(3)
Eg.
34 31
(3) 4 1
(5)3 (5)2
(5) 3 2
(-3)3
( 3) 3 2
( 3)1
(-3)
( 4) 4 2
( 4) 2
( 4)( 4)
(-3)2
( 4) 4
( 4) 2
Eg. (3)-4
(3)2
(3)
(3) 3
(5)1
4 2
(5)
(3)
6
1
(3) 3 (3)(3)(3)(3)
3
( 2) 2
( 2)( 2)
( 2) 1 ( 2) 3
( 2) 1
( 6) 3
(-6)-3-2 =
( 6) 5
1
( 6)( 6)( 6)( 6)( 6)
(2) 3
(2) 6
(2)(2)(2)(2)(2)(2)
(2)3
( 6) 2
(2)-3
c.
3
Complete the Quotient Law:
When dividing like bases, you must subtract the exponents.
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3.
Use the Power of a Power Law in the “Explore It” mode for the following
exercise.
Move the slider bars as directed:
m (purple) to 2
n (orange) to 3
x (red) to 4
a. Use the above example in the “Explore It” mode to complete the following:
Exponential Form
Expanded Form
(4 2 ) 3
(4 2 ) 3
(4) 2 3
(4 2 )(4 2 )(4 2 )
(4) 6
(4)(4) (4)(4) (4)(4)
(4) 6
b. Complete all the missing parts of the following table:
(xm )n
Exponential form
Expanded form(simplified)
eg. (23 )2
(2) 3 2
(5 2 ) 2
(5) 2 2
((-3)3)2
( 3) 3 2
( 3) 6
(-3)(-3)(-3)(-3)(-3)(-3)
((-4) 2 ) 2
( 4) 2 2
( 4) 4
(-4)(-4)(-4)(-4)
(2)(2)(2)(2)(2)(2)
(2) 6
(5) 4
( 3) 4 2
( 3) 8
((4)-3 )-2
(4) 3 2
(4) 6
((-2)-1 )-3
( 2) 1 3
((3)2 ) -2
(3) 2 2
e.g.(-3)-4)2
( 2) 3
(3) 4
(5)(5)(5)(5)
1
3
1
3
1
3
1
3
1
3
1
3
(4)(4)(4)(4)(4)(4)
(-2)(-2)(-2)
1
3
1
3
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1
3
1
3
Page 6 of 11
1
3
1
3
c. Complete the Power of a Power Law:
When raising a power to another power, you must multiply the exponents.
4.
Use the Power of a Product Law in the “Explore It” mode for the following
exercise.
Move the slider bars as directed:
m (purple) to 2
y (green) to 3
x (red) to 4
a. Use the above example in the “Explore It” mode to complete the following:
Exponential form
(4 3)
Expanded form
(4 3) 2
(4 3) (4 3)
(4)(4) (3)(3)
2
(4)1 2 (3)1 2
(4) 2 (3) 2
b. Complete all the missing parts in the following table:
(x y)m
Exponential form
Expanded form
e.g. (2 4)3
(2)
3
= (2) 3 (4)3
(2)(2)(2)×(4)(4)(4)
(3 2) 3
(3)1 3 (2)1 3
(3) 3 (2) 3
(3)(3)(3)×(2)(2)(2)
(-3 2)3
(-3) 1
-2
(4 3)
13
(4)1
3
(4) 1 2
(2)1 3 = ( 3) 3 (2) 3
(3) 1 2 = (4) 2 (3) 2
(-2 4) 3
(-2) 1 3
(4) 1 3 = (-2) 3
( 3 2) 2
( 3)1 2 (2)1 2
(4) 3
( 3) 2 (2) 2
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(-3)(-3)(-3)×(2)(2)(2)
1
4
1
2
1
4
1
3
1
2
1
3
1
2
1
4
(-3)(-3)×(2)(2)
Page 7 of 11
1
4
1
4
c. Complete the Power of a Product Law:
When finding the power of a product, you must multiply the exponent
outside the bracket by all the exponents inside the bracket.
5. Use the Power of a Quotient Law in the “Explore It” mode for the following
exercise.
Move the slider bars as directed:
m (purple) to 2
x (red) to 3
y (green) to 4
a. Use the above example in the “Explore It” mode to complete the following:
Exponential Form
3
4
Expanded Form
2
2
(3)1 2
3
4
(4)1 2
(3)1
(3)1
(3) 2
(4)1
(4)1
(4) 2
b. Complete all the missing parts in the following table:
x
y
2
4
Eg.
3
2
4
3
m
Exponential Form
(2)1 3
(2) 3
(4)1 3
(4) 3
31 4
(2)1 4
=
(3) 4
( 2) 4
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Expanded Form
2
4
3
2
1
1
2
4
3
2
1
1
2
4
3
2
1
1
3
2
Page 8 of 11
1
3
4
3
4
3
3
4
2
( 3)1 3
( 3) 3
(4)1 3
(4) 3
( 4)1 2
( 3)1 2
2
=
3
4
( 4) 2
4
3
( 3) 2
( 3)1 2
( 3) 2
( 4)1 2
( 4) 2
4
3
1
1
1
3
4
4
3
4
3
1
3
4
1
1
1
c. Complete the Power of a Quotient Law:
When finding the power of a quotient you must multiply the exponent
outside the bracket by all the exponents inside the bracket.
6.
Use the Zero Exponent Law in the “Explore It” mode for the following
exercise.
Move the slider bar as directed:
x (red) to 2
a. Use the above example in the “Explore It” mode to complete the following:
Exponential Form
20
1
Expanded Form
( 2) 0
2
2
1
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b. Complete all the missing parts in the following table:
(x)0
Exponential form
Expanded form
e.g. (4)0
(4)0 = 1
4
=1
4
(y)0
(y)0 = 1
y
y
1
(-3)
0
(-3) = 1
( 3)
( 3)
1
(-6)0
(-6)0 = 1
( 6)
( 6)
1
0
c. Complete the Zero Exponent Law:
The value of any base to the zero exponent is 1.
7.
Use the Negative Exponent Law in the “Explore It” mode for the following
exercise.
Move the slider bar as directed:
n (orange) to -1
x (red) to 2
.
a. Use the above example in the “Explore It” mode to complete the following:
Exponential form
(2)-1
1
2
Expanded form
(2)-1
1
( 2 )1
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b. Complete all the missing parts in the following table:
Exponential form
(x)-n
-2
e.g. (4)
1
(4)(4)
(5) 3
1
(5)(5)(5)
( 6)
2
1
( 6)( 6)
1
(4) 2
Expanded form
1
4
1
1
5
3
(5)
1
( 6)
2
1
6
1
4
1
5
1
5
1
6
c. Complete the Negative Exponent Law:
(x)-n is defined to be the reciprocal of (x)n.
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