188. Division with Power Expressions

Learning Intentions

• Recognise division expressions with like bases.
• Apply the division law for powers with variables.
• Simplify quotients involving positive integer indices.

Pre-requisite Summary

• Understand that the base is the repeated factor in a power, such as in .

• Understand that means .

• Know that powers can only be simplified Use the division law when the bases are the same.

• Know that when and .

• Know that coefficients are divided separately from powers.

• Know that different variables, such as and , must be simplified separately.

Worked Examples

Worked Example 1

Identify the like-bases in each division expression.

a)

b)

c)

Worked Example 2

State whether the division law for powers can be applied directly.

a)

b)

c)

Worked Example 3

Simplify each quotient.

a)

b)

c)

Worked Example 4

Simplify each quotient.

a)

b)

c)

Worked Example 5

Simplify each quotient.

a)

b)

c)

Worked Example 6

Simplify each expression.

a)

b)

c)

Problems

Problem 1

Identify the like-bases in each division expression.

a)

b)

c)

Problem 2

State whether the division law for powers can be applied directly.

a)

b)

c)

Problem 3

Simplify each quotient.

a)

b)

c)

Problem 4

Simplify each quotient.

a)

b)

c)

Problem 5

Simplify each quotient.

a)

b)

c)

Problem 6

Simplify each expression.

a)

b)

c)

Exercises

Understanding and Fluency

Exercise 1

Identify the like-bases in each expression.

a)

b)

c)

d)

Exercise 2

State whether the division law for powers can be applied directly.

a)

b)

c)

d)

Exercise 3

Simplify each quotient.

a)

b)

c)

d)

Exercise 4

Simplify each quotient.

a)

b)

c)

d)

Exercise 5

Simplify each quotient.

a)

b)

c)

d)

Exercise 6

Simplify each expression.

a)

b)

c)

d)

Exercise 7

Copy and complete each statement.

a)

b)

c)

d)

Exercise 8

Simplify each expression.

a)

b)

c)

d)

Exercise 9

Simplify each quotient.

a)

b)

c)

d)

Exercise 10

Simplify each expression.

a)

b)

c)

d)

Reasoning

Exercise 11

Explain why .

Exercise 12

A student writes . Explain the mistake.

Exercise 13

Explain why cannot be simplified using the division law for powers.

Exercise 14

A student writes . Explain the mistake.

Exercise 15

Decide whether each statement is true or false. Justify your answer.

a)

b)

c)

Problem-solving

Exercise 16

A rectangle has area and length .

a) Write an expression for the width.

b) Simplify the expression.

Exercise 17

A rectangle has area and length .

a) Write an expression for the width.

b) Simplify the expression.

Exercise 18

A prism has volume and base area .

a) Write an expression for the height.

b) Simplify the expression.

Exercise 19

A student divides by .

a) Write the quotient.

b) Simplify by dividing coefficients.

c) Simplify by applying the division law to each like-base.

Exercise 20

Create your own quotient of powers that simplifies to .

Your expression must include:

• coefficients in the numerator and denominator
• powers of both and
• numerator indices larger than denominator indices

Potential Misunderstandings

• Students may think the division law applies when the indices are the same, rather than when the bases are the same.
• Students may identify and as like-bases because they have the same index.
• Students may forget that means .
• Students may divide the indices instead of subtracting them, such as writing .
• Students may subtract in the wrong order, such as writing instead of at this level.
• Students may apply the division law to different bases, such as writing .
• Students may subtract coefficients instead of dividing them, such as writing .
• Students may simplify only one variable and leave the other variable unchanged.
• Students may forget that division by a variable expression assumes the variable is not equal to .

Next: 189. Powers Raised to Powers