142. Zero Indices and Powers of Powers

Learning Intentions

• To understand the meaning of an expression like
• Simplify expressions in which the index is zero
• simplify expressions involving powers of powers
• Expand expressions where a product is taken to a power, e.g.

Learning Intentions

• To understand the meaning of an expression like
• simplify expressions in which the index is zero
• simplify expressions involving powers of powers
• expand expressions where a product is taken to a power, e.g.

Pre-requisite Summary

• Know that an index tells how many times a base is multiplied by itself
• Be able to write powers such as in expanded form
• Know the index laws for multiplying and dividing terms with the same base
• Understand that a variable with no written index has index
• Be able to Identify factors inside brackets
• Know that repeated multiplication can be written more efficiently Use powers
• Understand that brackets group expressions together
• Be able to simplify simple algebraic products such as

Worked Examples

Worked Example 1

Write each expression in expanded form, then simplify:

a)

b)

c)

Worked Example 2

Simplify each expression with a zero index:

a)

b)

c)

Worked Example 3

Simplify each power of a power:

a)

b)

c)

Worked Example 4

Expand each product taken to a power:

a)

b)

c)

Worked Example 5

Expand and simplify:

a)

b)

c)

Worked Example 6

Simplify each expression:

a)

b)

c)

Problems

Problem 1

Write each expression in expanded form, then simplify:

a)

b)

c)

Problem 2

Simplify each expression with a zero index:

a)

b)

c)

Problem 3

Simplify each power of a power:

a)

b)

c)

Problem 4

Expand each product taken to a power:

a)

b)

c)

Problem 5

Expand and simplify:

a)

b)

c)

Problem 6

Simplify each expression:

a)

b)

c)

Problems

Understanding and Fluency

Exercise 1.

Write each expression in expanded form:

a)

b)

c)

Exercise 2.

Simplify each power of a power:

a)

b)

c)

d)

Exercise 3.

Simplify each expression with a zero index:

a)

b)

c)

d)

Exercise 4.

Simplify each expression:

a)

b)

c)

d)

Exercise 5.

Expand each product taken to a power:

a)

b)

c)

d)

Exercise 6.

Expand and 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)

Reasoning

Exercise 9.

Explain why means .

Exercise 10.

A student says that . Explain the mistake.

Exercise 11.

Noah says that because . Is he correct? Explain.

Exercise 12.

Explain why .

Exercise 13.

A student expands as . Describe the error.

Problem-solving

Exercise 14.

A pattern rule includes . Write this in simplest form.

Exercise 15.

A model for area uses . Expand and simplify the expression.

Exercise 16.

A science formula contains . Simplify the result.

Exercise 17.

A computer process gives . Simplify the output.

Exercise 18.

A student writes a term as . Simplify the term.

Exercise 19.

A design uses tiles in one section. Expand and simplify this expression.

Potential Misunderstandings

• Students may think means instead of multiplying the indices
• Students may think a zero index gives the value instead of
• Students may forget that the zero index rule applies only when the base is not zero
• Students may confuse a power of a power with multiplying terms that have the same base
• Students may add indices instead of multiplying them in expressions like
• Students may think means rather than
• Students may forget to Apply the power to the numerical coefficient as well, for example in
• Students may simplify as instead of
• Students may not Use brackets carefully and lose track of what the power applies to

Next: 143. Equivalent and Simplified Ratios