191. Powers of Quotients

Learning Intentions

• Recognise quotients raised to positive integer powers.
• Apply exponent laws to numerator and denominator terms.
• Simplify algebraic fractions involving powers.

Pre-requisite Summary

• Know that a quotient is the result of division, such as or .
• Know that a positive integer power means repeated multiplication, such as .
• Know that the outside index applies to both the numerator and the denominator.
• Know that , where .
• Know that the power of a power law is .
• Know that algebraic fractions can sometimes be simplified by applying index laws to like bases.

Worked Examples

Worked Example 1

Identify whether each expression shows a quotient raised to a positive integer power.

a)

b)

c)

Worked Example 2

Expand each expression by writing it as repeated multiplication.

a)

b)

c)

Worked Example 3

Simplify each expression.

a)

b)

c)

Worked Example 4

Simplify each expression.

a)

b)

c)

Worked Example 5

Simplify each expression.

a)

b)

c)

Worked Example 6

Simplify each algebraic fraction.

a)

b)

c)

Problems

Problem 1

Identify whether each expression shows a quotient raised to a positive integer power.

a)

b)

c)

Problem 2

Expand each expression by writing it as repeated multiplication.

a)

b)

c)

Problem 3

Simplify each expression.

a)

b)

c)

Problem 4

Simplify each expression.

a)

b)

c)

Problem 5

Simplify each expression.

a)

b)

c)

Problem 6

Simplify each algebraic fraction.

a)

b)

c)

Exercises

Understanding and Fluency

Exercise 1

Identify whether each expression shows a quotient raised to a positive integer power.

a)

b)

c)

d)

Exercise 2

Expand each expression by writing it as repeated multiplication.

a)

b)

c)

d)

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

Copy and complete each statement.

a)

b)

c)

d)

Exercise 9

Simplify each algebraic fraction.

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 .

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 length and width .

a) Write an expression for the area.

b) Express the area as a power.

c) Simplify the expression.

Exercise 17

A square has side length .

a) Write an expression for the area.

b) Simplify the expression.

Exercise 18

A cube has side length .

a) Write an expression for the volume.

b) Simplify the expression.

Exercise 19

A student simplifies .

a) Write the expression as repeated multiplication.

b) Simplify the coefficient.

c) Simplify the variable powers.

Exercise 20

Create your own expression that simplifies to .

Your expression must include:

• a quotient inside brackets
• a coefficient inside the numerator
• at least two variables
• an outside positive integer index

Potential Misunderstandings

• Students may not recognise as a quotient raised to a power because the power is outside the brackets.
• Students may confuse with .
• Students may forget that the denominator is also raised to the outside power.
• Students may apply the outside index to only the numerator, such as writing .
• Students may add indices instead of multiplying them, such as writing .
• Students may forget that a variable with no written index has index .
• Students may forget to raise coefficients to the outside power, such as writing instead of .
• Students may simplify algebraic fractions by cancelling terms before applying the outside power.
• Students may not State restrictions such as when a variable appears in the denominator.

Next: 192. Combining Index Laws