Algebraic indices - Teacher Slides.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Algebraic indices

Algebra

Lesson 1 of 7

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From the Number Chapter

Algebra

Lesson 1 of 7

Answer each one, then check.

1

Write 2⁵ × 2³ as a single power of 2.

2

Write (3²)⁴ as a single power of 3.

3

What is 5⁰?

4

What is 3⁻²?

5

Simplify 3a × 4b.

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From the Number Chapter - Answers

Algebra

Lesson 1 of 7

1

Write 2⁵ × 2³ as a single power of 2.

2⁸

2

Write (3²)⁴ as a single power of 3.

3⁸

3

What is 5⁰?

1

4

What is 3⁻²?

1/9

5

Simplify 3a × 4b.

12ab

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Learning Objectives

Algebra

Lesson 1 of 7

1

Use the index laws with letters.

2

Simplify expressions with numbers and several letters.

3

Raise a bracket to a power, such as (2x³)⁴.

4

Use negative and fractional indices in algebra. (Higher)

5

Solve equations with unknown powers, such as 9^x = 27^(x−1). (Higher)

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The Index Laws with Letters

Algebra

Lesson 1 of 7

Exactly the laws from the Number chapter - the base is now a letter.

Law

In words

Example

Multiplying

Add the indices

x⁵ × x² = x⁷

Dividing

Subtract the indices

y⁸ ÷ y² = y⁶

Power of a power

Multiply the indices

(z³)⁵ = z¹⁵

Zero index

Anything (except 0) to the power 0 is 1

x⁰ = 1

A letter on its own

Has an index of 1

x × x⁴ = x¹ × x⁴ = x⁵

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Multiplying Terms

Algebra

Lesson 1 of 7

Simplify 3x²y⁴ × 5x³y

1

Multiply the numbers

3 × 5 = 15

2

Multiply the x terms: add the indices

x² × x³ = x⁵

3

Multiply the y terms: y is y¹

y⁴ × y¹ = y⁵

4

Put them together

15x⁵y⁵

ANSWER

15x⁵y⁵

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Dividing Terms

Algebra

Lesson 1 of 7

Simplify 24a⁷b³ ÷ 8a²b³

1

Divide the numbers

24 ÷ 8 = 3

2

Divide the a terms: subtract the indices

a⁷ ÷ a² = a⁵

3

Divide the b terms

b³ ÷ b³ = b⁰ = 1

4

Put them together

3 × a⁵ × 1

ANSWER

3a⁵

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A Bracket to a Power

Algebra

Lesson 1 of 7

Simplify (3x²y⁵)³

1

Everything inside the bracket is raised to the power 3

3³ × (x²)³ × (y⁵)³

2

The number

3³ = 27

3

Power of a power: multiply the indices

(x²)³ = x⁶ and (y⁵)³ = y¹⁵

ANSWER

27x⁶y¹⁵

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Why a Bracket Cubed Cubes the Number Too

Algebra

Lesson 1 of 7

(2x)³ = 2x × 2x × 2x = 8x³, not 2x³.

The power on a bracket applies to everything inside it - the number as well as the letter. A cube with edges of 2x has a volume of 2x × 2x × 2x = 8x³. Writing 2x³ would mean only the x was cubed.

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Mistakes to Avoid

Algebra

Lesson 1 of 7

WRONG

• (2x)³ = 2x³

• 3x² × 2x = 6x²

• x² + x³ = x⁵

• (x³)² = x⁵

• 12x⁶ ÷ 4x² = 3x³

RIGHT

• (2x)³ = 8x³ - cube the 2 as well.

• 3x² × 2x = 6x³ - x is x¹.

• x² + x³ cannot be simplified: they are not like terms.

• (x³)² = x⁶ - multiply the indices.

• 12x⁶ ÷ 4x² = 3x⁴ - subtract the indices.

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PART TWO · HIGHER

Negative and Fractional Indices

The rest of the index rules, now with letters.

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Negative and Fractional Powers of Letters

Algebra

Lesson 1 of 7

The meanings are the same as for numbers.

Negative powers

x⁻¹ = 1/x and x⁻³ = 1/x³. So 1/x⁴ can be written as x⁻⁴.

Fractional powers

x^(1/2) = √x and x^(1/3) = ∛x. So 1/(√x) = x^(−1/2).

"Write in the form xⁿ"

Turn roots and fractions into single powers: 1/x² = x⁻², √x³ = x^(3/2).

Brackets

Raise each part inside the bracket to the power: (16x⁸)^(1/2) = 16^(1/2) × x⁴ = 4x⁴.

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A Negative Fractional Power

Algebra

Lesson 1 of 7

Simplify (27x⁶y⁻³)^(−1/3)

1

Raise each part to the power −⅓

27^(−1/3) × (x⁶)^(−1/3) × (y⁻³)^(−1/3)

2

The number: flip and cube root

27^(−1/3) = ⅓

3

The x: multiply the indices

6 × (−⅓) = −2, so x⁻² = 1/x²

4

The y: multiply the indices

−3 × (−⅓) = 1, so y¹ = y

5

Put them together

⅓ × 1/x² × y

ANSWER

y/3x²

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Solving an Equation with Powers

Algebra

Lesson 1 of 7

Solve 2^(x+1) = 8^x

1

Write both sides with the same base

8 = 2³, so 8^x = 2^(3x)

2

So

2^(x+1) = 2^(3x)

3

Same base, so the indices are equal

x + 1 = 3x

4

Solve

1 = 2x

ANSWER

x = ½

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Key Terms

Algebra

Lesson 1 of 7

Index (power)

The small number showing how many times the base is multiplied by itself.

Coefficient

The number in front of a letter, e.g. 5 in 5x³.

Like terms

Terms with exactly the same letters and powers, e.g. 3x² and 7x².

Simplify

Write an expression in its shortest form.

Negative index (Higher)

x⁻ⁿ = 1/xⁿ.

Fractional index (Higher)

x^(1/n) = ⁿ√x.

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Algebra

Lesson 1 of 7

YOUR TASK

Index Pyramids

12 minutes

Simplify each. (a) a⁴ × a⁶ (b) 6m⁵ × 2m (c) 18p⁸q³ ÷ 6p²q (d) (4x³)² (e) (2a²b)³ × 3ab⁴ (f) (x⁷ × x²)/x⁵. Higher: (g) (9x⁴)^(1/2) (h) (8y⁶)^(−1/3) (i) solve 4^x = 2^(x+3).

1

Numbers first.

2

Then each letter in turn.

3

Check every bracket's power applies to everything inside.

WHAT A GOOD ANSWER SHOWS

(a) a¹⁰ (b) 12m⁶ (c) 3p⁶q² (d) 16x⁶ (e) 8a⁶b³ × 3ab⁴ = 24a⁷b⁷ (f) x⁴ (g) 3x² (h) 1/2y² (i) 2^(2x) = 2^(x+3), so x = 3.

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Can I...?

Algebra

Lesson 1 of 7

Multiply terms using the index laws.

Divide terms using the index laws.

Raise a power to a power.

Raise a bracket to a power.

Use x⁰ = 1.

Use negative powers of letters. (Higher)

Use fractional powers of letters. (Higher)

Solve equations with unknown powers. (Higher)

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Summary & Exam Focus

Multiply: add indices. Divide: subtract indices. Power of a power: multiply indices.

A power on a bracket applies to everything inside, including the number.

Only like terms can be added; x² + x³ does not simplify.

(Higher) x⁻ⁿ = 1/xⁿ and x^(1/n) = ⁿ√x.

EXAM FOCUS

Simplify (2a³b)⁵ (2 marks)

Work through the numbers first, then each letter in alphabetical order. The two marks are usually one for the number and one for the letters.

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Exam Practice: Algebraic Indices

Algebra

Lesson 1 of 7

Answer all questions. Show your working. All questions are non-calculator. · 20 minutes

Question 1 · 1 mark · Non-calculator

Simplify m⁵ × m³

Question 2 · 1 mark · Non-calculator

Simplify (p⁴)³

Question 3 · 2 marks · Non-calculator

Simplify 4x²y³ × 3x⁵y

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Exam Practice: Algebraic Indices (cont.)

Algebra

Lesson 1 of 7

Question 4 · 2 marks · Non-calculator

Simplify (2a³b)⁵

Question 5 · 2 marks · Non-calculator · Higher

Simplify (64x⁶)^(2/3)

Question 6 · 3 marks · Non-calculator · Higher

Solve 9^x = 27^(x−1)

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Question 1 · 1 mark · Non-calculator

Algebra

Lesson 1 of 7

“

Simplify m⁵ × m³

HOW TO ANSWER IT

Command word: Non-calculator. Worth 1 mark, so plan before writing.

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Question 1 · mark scheme

Algebra

Lesson 1 of 7

1 mark available. Award a mark for each point made.

m⁸

B1

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Question 2 · 1 mark · Non-calculator

Algebra

Lesson 1 of 7

“

Simplify (p⁴)³

HOW TO ANSWER IT

Command word: Non-calculator. Worth 1 mark, so plan before writing.

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Question 2 · mark scheme

Algebra

Lesson 1 of 7

1 mark available. Award a mark for each point made.

p¹²

B1

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Question 3 · 2 marks · Non-calculator

Algebra

Lesson 1 of 7

“

Simplify 4x²y³ × 3x⁵y

HOW TO ANSWER IT

Command word: Non-calculator. Worth 2 marks, so plan before writing.

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Question 3 · mark scheme

Algebra

Lesson 1 of 7

2 marks available. Award a mark for each point made.

12x⁷y⁴

B2

Two of 12, x⁷ and y⁴ in a single term

B1

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Question 4 · 2 marks · Non-calculator

Algebra

Lesson 1 of 7

“

Simplify (2a³b)⁵

HOW TO ANSWER IT

Command word: Non-calculator. Worth 2 marks, so plan before writing.

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Question 4 · mark scheme

Algebra

Lesson 1 of 7

2 marks available. Award a mark for each point made.

32a¹⁵b⁵

B2

Two of 32, a¹⁵ and b⁵ in a single term

B1

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Question 5 · 2 marks · Non-calculator · Higher

Algebra

Lesson 1 of 7

“

Simplify (64x⁶)^(2/3)

HOW TO ANSWER IT

Command word: Non-calculator · Higher. Worth 2 marks, so plan before writing.

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Question 5 · mark scheme

Algebra

Lesson 1 of 7

2 marks available. Award a mark for each point made.

16x⁴

B2

16 or x⁴ in a single term

B1

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Question 6 · 3 marks · Non-calculator · Higher

Algebra

Lesson 1 of 7

“

Solve 9^x = 27^(x−1)

HOW TO ANSWER IT

Command word: Non-calculator · Higher. Worth 3 marks, so plan before writing.

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Question 6 · mark scheme

Algebra

Lesson 1 of 7

3 marks available. Award a mark for each point made.

Both sides written as powers of 3

M1

2x = 3x − 3

M1

x = 3

A1

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