141. Indices and Index Laws

Indices and Index Laws

Learning Intentions

• To understand the meaning of an expression in the form in terms of repeated multiplication of
• To know the meaning of the terms base, index (plural indices) and expanded form
• Apply the index laws for multiplying terms with the same base
• apply the index laws for dividing terms with the same base

Pre-requisite Summary

• Know that multiplication can be written as repeated addition in simple cases
• Understand that repeated multiplication of the same factor can be written more efficiently
• Be able to multiply whole numbers and algebraic terms
• Know that a term can contain numbers and pronumerals
• Be able to Simplify simple algebraic products such as
• Understand that a fraction bar can represent division
• Be able to Identify factors in a product

Worked Examples

Worked Example 1

Write each expression in expanded form and State its meaning:

a)

b)

c)

Worked Example 2

Identify the base and index in each expression:

a)

b)

c)

Worked Example 3

Use the index law for multiplication with the same base:

a)

b)

c)

Worked Example 4

Use the index law for multiplication with the same base:

a)

b)

c)

Worked Example 5

Use the index law for division with the same base:

a)

b)

c)

Worked Example 6

Use the index laws to simplify:

a)

b)

c)

Problems

Problem 1

Write each expression in expanded form and state its meaning:

a)

b)

c)

Problem 2

Identify the base and index in each expression:

a)

b)

c)

Problem 3

Use the index law for multiplication with the same base:

a)

b)

c)

Problem 4

Use the index law for multiplication with the same base:

a)

b)

c)

Problem 5

Use the index law for division with the same base:

a)

b)

c)

Problem 6

Use the index laws to simplify:

a)

b)

c)

Exercises

Understanding and Fluency

Exercise 1.

Write each expression in expanded form:

a)

b)

c)

d)

Exercise 2.

State the base and the index in each expression:

a)

b)

c)

d)

Exercise 3.

Complete each statement:

a) In , the base is ______

b) In , the index tells how many times ______ is multiplied by itself

c) The expanded form of is ______

Exercise 4.

Use the index law for multiplication with the same base:

a)

b)

c)

d)

Exercise 5.

Use the index law for multiplication with the same base:

a)

b)

c)

d)

Exercise 6.

Use the index law for division with the same base:

a)

b)

c)

d)

Exercise 7.

Use the index law for division with the same base:

a)

b)

c)

d)

Exercise 8.

Simplify each expression Use the index laws:

a)

b)

c)

d)

Reasoning

Exercise 9.

Explain why means , and not .

Exercise 10.

A student says that in , the base is and the index is . Explain the mistake.

Exercise 11.

Noah says that because . Is he correct? Explain.

Exercise 12.

Explain why .

Exercise 13.

A student says that because . Describe the error.

Problem-solving

Exercise 14.

A square has side length cm and another factor of is multiplied into a calculation for a pattern. Simplify .

Exercise 15.

A science formula includes . Simplify the expression.

Exercise 16.

A computer process multiplies by and then divides by . Simplify the result.

Exercise 17.

A student writes the expanded form of to help Check a pattern rule. Write the expanded form and then simplify .

Exercise 18.

A design uses tiles in one section. Write the simplified power of .

Exercise 19.

A quantity is modelled by . Simplify the model.

Potential Misunderstandings

• Students may think means instead of repeated multiplication of
• Students may confuse the base with the index
• Students may think the index tells the value of the term rather than the number of repeated factors
• Students may write the expanded form incorrectly by using addition instead of multiplication
• Students may multiply indices when multiplying terms with the same base
• Students may divide indices when dividing terms with the same base
• Students may apply the index laws when the bases are different
• Students may forget that a variable with no written index has index