192. Combining Index Laws
Learning Intentions
- Choose appropriate exponent laws for multi-step expressions.
- Simplify algebraic expressions involving several index laws.
- Check simplified expressions by comparing equivalent forms.
Pre-requisite Summary
- Know the multiplication law for powers with like bases:
. - Know the division law for powers with like bases:
when and . - Know the power of a power law:
. - Know the power of a product law:
. - Know the power of a quotient law:
, where . - Know that equivalent algebraic expressions have the same value for every allowed value of the variable.
Worked Examples
Worked Example 1
Select the index law that should be applied first.
a)
b)
c)
Worked Example 2
Select the index laws needed to simplify each expression.
a)
b)
c)
Worked Example 3
Simplify each expression.
a)
b)
c)
Worked Example 4
Simplify each expression.
a)
b)
c)
Worked Example 5
Check whether the two expressions are equivalent.
a)
b)
c)
Worked Example 6
Simplify the expression, then check by substituting
Problems
Problem 1
Select the index law that should be applied first.
a)
b)
c)
Problem 2
Select the index laws needed to simplify each expression.
a)
b)
c)
Problem 3
Simplify each expression.
a)
b)
c)
Problem 4
Simplify each expression.
a)
b)
c)
Problem 5
Check whether the two expressions are equivalent.
a)
b)
c)
Problem 6
Simplify the expression, then check by substituting
Exercises
Understanding and Fluency
Exercise 1
Select the index law that should be applied first.
a)
b)
c)
d)
Exercise 2
Match each expression to the main index law needed first.
a)
b)
c)
d)
Laws:
- multiplication law
- division law
- power of a power law
- power of a product law
Exercise 3
Simplify each expression.
a)
b)
c)
d)
Exercise 4
Simplify each expression.
a)
b)
c)
d)
Exercise 5
Simplify each expression.
a)
b)
c)
d)
Exercise 6
Simplify each expression.
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
Check whether each pair of expressions is equivalent.
a)
b)
c)
d)
Reasoning
Exercise 11
Explain why
Exercise 12
A student writes
Exercise 13
A student writes
Exercise 14
Explain why
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.
c) Check your expression by substituting
Exercise 17
A rectangle has area
a) Write an expression for the width.
b) Simplify the expression.
c) Check your expression by multiplying the length and width.
Exercise 18
A square has side length
a) Write an expression for the area.
b) Simplify the expression.
c) Explain which index laws were used.
Exercise 19
A prism has volume
a) Write an expression for the height.
b) Simplify the expression.
c) Check that your height multiplied by the base area gives the original volume.
Exercise 20
Create your own multi-step index-law expression that simplifies to
Your expression must include:
- at least two different index laws
- a coefficient
- both variables
and - a way to check the simplified expression
Potential Misunderstandings
- Students may Choose the wrong law because they focus on the indices rather than the operation between terms.
- Students may Apply the multiplication law to powers raised to powers, such as treating
like . - Students may not Recognise when more than one index law is needed.
- Students may add indices when they should multiply them, such as
. - Students may multiply indices when they should add them, such as
. - Students may subtract indices before expanding brackets, leading to errors in expressions such as
. - Students may think checking means repeating the same simplification rather than comparing equivalent forms.
- Students may check Use only one value and assume that this proves equivalence for all values.
- Students may forget that expressions involving denominators require restrictions such as
or .