189. Powers Raised to Powers

Learning Intentions

• Identify expressions involving powers raised to another power.
• Apply the power of a power law to variable expressions.
• Simplify algebraic powers Use index laws.

Pre-requisite Summary

• Know that the base is the repeated factor in a power, such as in .
• Know that the index tells how many times the base is used as a factor.
• Know that a power raised to another power has the form .
• Know that the power of a power law is for positive integer indices.
• Know that the multiplication law for powers is .
• Know that each variable base in an algebraic expression must be simplified separately.

Worked Examples

Worked Example 1

Identify whether each expression shows a power raised to another power.

a)

b)

c)

Worked Example 2

Identify the base, inside index and outside index in 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

Simplify each expression.

a)

b)

c)

Worked Example 6

Simplify each expression using index laws.

a)

b)

c)

Problems

Problem 1

Identify whether each expression shows a power raised to another power.

a)

b)

c)

Problem 2

Identify the base, inside index and outside index in each expression.

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 using index laws.

a)

b)

c)

Exercises

Understanding and Fluency

Exercise 1

Identify whether each expression shows a power raised to another power.

a)

b)

c)

d)

Exercise 2

Identify the base, inside index and outside index in each expression.

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 using index laws.

a)

b)

c)

d)

Exercise 8

Simplify each expression using index laws.

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 the difference between and .

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 cube has side length .

a) Write an expression for the volume.

b) Simplify the expression.

Exercise 18

A rectangular prism has side lengths , and .

a) Write an expression for the volume.

b) Simplify the expression using index laws.

Exercise 19

A student simplifies .

a) Write the expression as a product of factors.

b) Simplify the coefficient.

c) Simplify the variable powers.

Exercise 20

Create your own expression that simplifies to .

Your expression must include:

• a power raised to another power
• both variables and
• positive integer indices

Potential Misunderstandings

• Students may not Recognise as a power raised to another power because there are two indices.
• Students may confuse the inside index and outside index.
• Students may think and are simplified in the same way.
• Students may add the indices instead of multiplying them, such as writing .
• Students may multiply only one variable index in expressions such as .
• Students may forget that means in expressions such as .
• Students may apply the power of a power law before noticing other operations, such as multiplication or division by another like-base.
• Students may confuse the multiplication law with the power of a power law .
• Students may simplify coefficients incorrectly when the whole expression is raised to a power, such as writing instead of .

Next: 190. Power of Products