194. Identifying Errors in Index Expressions

Learning Intentions

• Identify common errors when applying exponent laws.
• Correct incorrect simplifications involving powers.
• Justify correct Use of exponent laws with examples.

Pre-requisite Summary

• Know the multiplication law for like bases: .
• Know the division law for like bases: , where and .
• Know the power of a power law: .
• Know the power of a product law: .
• Know that exponent laws depend on the operation being used.
• Know that a correct justification should name the index law and Describe how the indices change.

Worked Examples

Worked Example 1

Identify the error in each incorrect simplification.

a)

b)

c)

Worked Example 2

Identify the error in each incorrect simplification.

a)

b)

c)

Worked Example 3

Correct each simplification.

a)

b)

c)

Worked Example 4

Correct each simplification.

a)

b)

c)

Worked Example 5

Decide whether each simplification is correct. If it is incorrect, correct it.

a)

b)

c)

Worked Example 6

Justify the correct simplification Use index law language.

Worked Example 7

Justify the correct simplification using index law language.

Worked Example 8

Write an example that shows the difference between the multiplication law and the power of a power law.

Problems

Problem 1

Identify the error in each incorrect simplification.

a)

b)

c)

Problem 2

Identify the error in each incorrect simplification.

a)

b)

c)

Problem 3

Correct each simplification.

a)

b)

c)

Problem 4

Correct each simplification.

a)

b)

c)

Problem 5

Decide whether each simplification is correct. If it is incorrect, correct it.

a)

b)

c)

Problem 6

Justify the correct simplification using index law language.

Problem 7

Justify the correct simplification using index law language.

Problem 8

Write an example that shows the difference between the division law and the power of a power law.

Exercises

Understanding and Fluency

Exercise 1

Identify the error in each incorrect simplification.

a)

b)

c)

d)

Exercise 2

Identify the error in each incorrect simplification.

a)

b)

c)

d)

Exercise 3

Identify the error in each incorrect simplification.

a)

b)

c)

d)

Exercise 4

Identify the error in each incorrect simplification.

a)

b)

c)

d)

Exercise 5

Correct each simplification.

a)

b)

c)

d)

Exercise 6

Correct each simplification.

a)

b)

c)

d)

Exercise 7

Correct each simplification.

a)

b)

c)

d)

Exercise 8

Correct each simplification.

a)

b)

c)

d)

Exercise 9

Decide whether each simplification is correct. If it is incorrect, correct it.

a)

b)

c)

d)

Exercise 10

Decide whether each simplification is correct. If it is incorrect, correct it.

a)

b)

c)

d)

Reasoning

Exercise 11

A student writes:

Explain the mistake and give the correct simplification.

Exercise 12

A student writes:

Explain the mistake and give the correct simplification.

Exercise 13

A student writes:

Explain the mistake and give the correct simplification.

Exercise 14

A student writes:

Explain the mistake and give the correct simplification.

Exercise 15

Use examples to explain the difference between:

a)

b)

Exercise 16

Use examples to explain why , not .

Problem-solving

Exercise 17

A student simplified the area of a rectangle incorrectly.

Length:

Width:

Student answer:

a) Identify the error.

b) Correct the simplification.

c) Justify the answer using index law language.

Exercise 18

A student simplified the width of a rectangle incorrectly.

Area:

Length:

Student answer:

a) Identify the error.

b) Correct the simplification.

c) Justify the answer using index law language.

Exercise 19

A student simplified the volume of a cube incorrectly.

Side length:

Student answer:

a) Identify the error.

b) Correct the simplification.

c) Explain why the coefficient must also be raised to the power.

Exercise 20

Create your own incorrect simplification involving powers.

Your response must include:

• the incorrect simplification
• the error that was made
• the correct simplification
• a justification using index law language

Potential Misunderstandings

• Students may multiply indices when multiplying like bases, such as writing .
• Students may divide indices when dividing like bases, such as writing .
• Students may add indices when raising a power to a power, such as writing .
• Students may Apply an index law to different bases, such as writing .
• Students may apply the outside index to only one factor in a product, such as writing .
• Students may apply the outside index to only the numerator in a quotient, such as writing .
• Students may forget to raise coefficients to the outside power, such as writing .
• Students may give the correct simplified expression without being able to name the index law used.
• Students may use vague explanations such as “the powers cancel” instead of explaining the operation on the indices.