187r. Multiplying Powers and Variables
Learning Intentions
- Identify like-bases in algebraic expressions.
- Apply the multiplication law for powers with variables.
- Simplify products of powers Use positive integer indices.
Pre-requisite Summary
- Understand that the base is the repeated factor in a power, such as
in . - Understand that the index tells how many times the base is multiplied by itself.
- Know that powers can only be combined using the multiplication law when the bases are the same.
- Know that
for positive integer indices. - Know that coefficients are multiplied separately from powers.
- Know that different variables, such as
and , are not like-bases.
Worked Examples
Worked Example 1
Identify the like-bases in each expression.
a)
b)
c)
Worked Example 2
State whether the multiplication law for powers can be applied directly.
a)
b)
c)
Worked Example 3
Simplify each product.
a)
b)
c)
Worked Example 4
Simplify each product.
a)
b)
c)
Worked Example 5
Simplify each product.
a)
b)
c)
Worked Example 6
Simplify each expression.
a)
b)
c)
Problems
Problem 1
Identify the like-bases in each expression.
a)
b)
c)
Problem 2
State whether the multiplication law for powers can be applied directly.
a)
b)
c)
Problem 3
Simplify each product.
a)
b)
c)
Problem 4
Simplify each product.
a)
b)
c)
Problem 5
Simplify each product.
a)
b)
c)
Problem 6
Simplify each expression.
a)
b)
c)
Exercises
Understanding and Fluency
Exercise 1.
Identify the base and index in each power.
a)
b)
c)
d)
Exercise 2.
Identify the like-bases in each expression.
a)
b)
c)
d)
Exercise 3.
State whether the multiplication law for powers can be applied directly.
a)
b)
c)
d)
Exercise 4.
Simplify each product.
a)
b)
c)
d)
Exercise 5.
Simplify each product.
a)
b)
c)
d)
Exercise 6.
Simplify each product.
a)
b)
c)
d)
Exercise 7.
Simplify each expression.
a)
b)
c)
d)
Exercise 8.
Simplify each expression.
a)
b)
c)
d)
Exercise 9.
Copy and complete each statement.
a)
b)
c)
d)
Exercise 10.
Simplify each product.
a)
b)
c)
d)
Reasoning
Exercise 11.
Explain why
Exercise 12.
A student writes
Exercise 13.
Explain why
Exercise 14.
A student writes
Exercise 15.
Decide whether each statement is true or false. Justify your answer.
a)
b)
c)
Problem-solving
Exercise 16.
A rectangle has length
a) Write an expression for the area.
b) Simplify the expression.
Exercise 17.
A rectangle has length
a) Write an expression for the area.
b) Simplify the expression.
Exercise 18.
A cuboid has side lengths
a) Write an expression for the volume.
b) Simplify the expression.
Exercise 19.
A student multiplies
a) Write the product.
b) Simplify by multiplying coefficients.
c) Simplify by applying the multiplication law to like-bases.
Exercise 20.
Create your own product of powers that simplifies to
Your expression must include:
- two factors
- coefficients in both factors
- powers of both
and
Potential Misunderstandings
- Students may think the index is the base, rather than the small raised number.
- Students may identify terms as like-bases because the indices are the same, such as
and . - Students may forget that
means . - Students may multiply indices instead of adding them, such as writing
. - Students may apply the multiplication law when the bases are different, such as writing
. - Students may combine coefficients and indices incorrectly.
- Students may add coefficients instead of multiplying them, such as writing
. - Students may leave products partly simplified by combining only one variable.
- Students may forget that each variable base must be simplified separately in expressions such as
.