190. Power of Products

Learning Intentions

• Expand powers of products involving variables.
• Apply exponent laws to each factor in a product.
• Simplify expressions such as , and .

Pre-requisite Summary

• Know that a power means repeated multiplication, such as .
• Know that a product is the result of multiplying factors together, such as or .
• Know that the power of a power law is .
• Know that when a product is raised to a power, the outside index applies to every factor inside the brackets.
• Know that coefficients are raised to powers as well as variables.
• Know that variables with no written index have an index of , such as .

Worked Examples

Worked Example 1

Expand each power of a product.

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 expression.

a)

b)

c)

Problems

Problem 1

Expand each power of a product.

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 expression.

a)

b)

c)

Exercises

Understanding and Fluency

Exercise 1

Expand each power of a product.

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

Simplify each expression.

a)

b)

c)

d)

Exercise 9

Copy and complete each statement.

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 square has side length .

a) Write an expression for the area.

b) 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 rectangular prism has side lengths , and .

a) Write an expression for the volume.

b) Express the volume as a power.

c) Simplify the expression.

Exercise 20

Create your own expression that simplifies to .

Your expression must include:

• a product inside brackets
• a coefficient inside brackets
• two variables
• an outside index

Potential Misunderstandings

• Students may apply the outside index to only one factor, such as writing .
• Students may forget that variables without written indices have an index of .
• Students may not Recognise that means .
• Students may add indices instead of multiplying them, such as writing .
• Students may multiply only the first variable index and leave the other unchanged.
• Students may confuse the product law with the power of a product law .
• Students may forget to raise the coefficient to the outside power, such as writing instead of .
• Students may treat as instead of applying the power to every factor.
• Students may simplify the variable powers correctly but make arithmetic errors with coefficients.

Next: 191. Powers of Quotients