193. Index Laws in Algebraic Problems

Learning Intentions

• Apply exponent laws to Solve simple algebraic problems.
• Interpret variables and powers in mathematical statements.
• Explain each simplification step Use index law language.

Pre-requisite Summary

• Know the multiplication law for like bases: .
• Know the division law for like bases: , where and .
• Know that variables can represent unknown numbers or changing quantities.
• Know that powers can represent repeated multiplication, area, volume or repeated scaling.
• Know the power of a power law: .
• Know how to Describe index laws using words such as base, index, product, quotient and power.

Worked Examples

Worked Example 1

A rectangle has length and width .

a) Write an expression for the area.

b) Simplify the expression using an index law.

c) State the index law used.

Worked Example 2

A rectangle has area and length .

a) Write an expression for the width.

b) Simplify the expression using an index law.

c) State the index law used.

Worked Example 3

A square has side length .

a) Write an expression for the area.

b) Simplify the expression using an index law.

c) Explain why a power of a power is involved.

Worked Example 4

In the expression , the variable represents the side length of a small square tile.

Explain what could mean if it is written as .

Worked Example 5

A cube has side length .

a) Write an expression for the volume.

b) Simplify the expression.

c) Interpret what the coefficient and index represent.

Worked Example 6

Simplify the expression and explain each step using index law language.

Worked Example 7

Simplify the expression and explain each step using index law language.

Worked Example 8

Check whether the following mathematical statement is true.

Explain your reasoning using index law language.

Problems

Problem 1

A rectangle has length and width .

a) Write an expression for the area.

b) Simplify the expression using an index law.

c) State the index law used.

Problem 2

A rectangle has area and length .

a) Write an expression for the width.

b) Simplify the expression using an index law.

c) State the index law used.

Problem 3

A square has side length .

a) Write an expression for the area.

b) Simplify the expression using an index law.

c) Explain why a power of a power is involved.

Problem 4

In the expression , the variable represents a length scale.

Explain what could mean if it is written as .

Problem 5

A cube has side length .

a) Write an expression for the volume.

b) Simplify the expression.

c) Interpret what the coefficient and index represent.

Problem 6

Simplify the expression and explain each step using index law language.

Problem 7

Simplify the expression and explain each step using index law language.

Problem 8

Check whether the following mathematical statement is true.

Explain your reasoning using index law language.

Exercises

Understanding and Fluency

Exercise 1

For each situation, write and simplify an algebraic expression.

a) A rectangle has length and width . Solve the area.

b) A rectangle has length and width . Find the area.

c) A rectangle has length and width . Find the area.

Exercise 2

For each situation, write and simplify an algebraic expression.

a) A rectangle has area and length . Find the width.

b) A rectangle has area and length . Find the width.

c) A rectangle has area and length . Find the width.

Exercise 3

For each square, write and simplify an expression for the area.

a) Side length

b) Side length

c) Side length

Exercise 4

For each cube, write and simplify an expression for the volume.

a) Side length

b) Side length

c) Side length

Exercise 5

Simplify each expression and name the first index law used.

a)

b)

c)

d)

Exercise 6

Simplify each expression and write one sentence explaining the index law used.

a)

b)

c)

d)

Exercise 7

Interpret each algebraic statement in words.

a) means is used as a factor three times.

b)

c)

d)

Exercise 8

Simplify each expression.

a)

b)

c)

d)

Exercise 9

Simplify each expression.

a)

b)

c)

d)

Exercise 10

Copy and complete each explanation.

a) because when multiplying powers with the same base, we WHAT the indices.

b) because when dividing powers with the same base, we WHAT the indices.

c) because when raising a power to a power, we WHAT the indices.

Reasoning

Exercise 11

Explain why a rectangle with length and width has area .

Exercise 12

A student writes:

Explain the mistake using index law language.

Exercise 13

A student writes:

Explain the mistake using index law language.

Exercise 14

A student writes:

Explain the mistake using index law language.

Exercise 15

Decide whether each mathematical statement is true or false. Justify your answer.

a)

b)

c)

Problem-solving

Exercise 16

A rectangular garden has length metres and width metres.

a) Write an expression for the area.

b) Simplify the expression.

c) Explain the simplification using index law language.

Exercise 17

A rectangular sign has area square centimetres and length centimetres.

a) Write an expression for the width.

b) Simplify the expression.

c) Explain which index law was used.

Exercise 18

A cube has side length centimetres.

a) Write an expression for the volume.

b) Simplify the expression.

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

Exercise 19

A prism has volume cubic units and base area square units.

a) Write an expression for the height.

b) Simplify the expression.

c) Check that your height multiplied by the base area gives the original volume.

Exercise 20

Create your own algebraic problem that simplifies to .

Your problem must include:

• a real or mathematical context
• at least one index law
• a simplified answer
• one sentence explaining the index law used

Potential Misunderstandings

• Students may Use the wrong index law because they focus on the powers instead of the operation connecting the terms.
• Students may not Recognise that area problems often involve multiplying powers, while missing-side problems often involve dividing powers.
• Students may write an expression correctly but fail to simplify it using index laws.
• Students may interpret as rather than multiplied by itself five times.
• Students may forget that variables represent values, so expressions such as can represent lengths, areas, volumes or scaling relationships depending on context.
• Students may not distinguish between the meaning of the coefficient and the meaning of the index.
• Students may explain simplification steps vaguely, such as saying “I just simplified it,” instead of naming the index law used.
• Students may say “multiply the powers” when they mean “add the indices” for like-base multiplication.
• Students may say “cancel the powers” instead of explaining that the division law subtracts indices for like bases.

Next: 194. Identifying Errors in Index Expressions