Zero negative and fractional indices.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Zero, negative and fractional indices

Number

Lesson 5 of 7

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Last Lesson

Number

Lesson 5 of 7

Answer each one, then check.

1

Write 2⁵ × 2³ as a single power of 2.

2

Write 3⁷ ÷ 3² as a single power of 3.

3

Write (5²)³ as a single power of 5.

4

What is ∛27?

5

What is the reciprocal of 4?

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Last Lesson - Answers

Number

Lesson 5 of 7

1

Write 2⁵ × 2³ as a single power of 2.

2⁸

2

Write 3⁷ ÷ 3² as a single power of 3.

3⁵

3

Write (5²)³ as a single power of 5.

5⁶

4

What is ∛27?

3

5

What is the reciprocal of 4?

¼ - one divided by the number.

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Learning Objectives

Number

Lesson 5 of 7

1

Know that any number (except 0) to the power 0 is 1.

2

Work out negative powers as reciprocals.

3

Work out negative powers of fractions.

4

Work out fractional powers as roots. (Higher)

5

Work out powers such as 27^(−2/3). (Higher)

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Follow the Pattern

Number

Lesson 5 of 7

Each step down, the index goes down by 1 and the value is divided by 2.

Power of 2

Value

How it continues

2³

8

2²

4

8 ÷ 2

2¹

2

4 ÷ 2

2⁰

1

2 ÷ 2

2⁻¹

½

1 ÷ 2

2⁻²

¼

½ ÷ 2

2⁻³

⅛

¼ ÷ 2

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The Zero Index

Number

Lesson 5 of 7

Any number except 0 raised to the power 0 is 1.

Using the laws

5³ ÷ 5³ = 5³⁻³ = 5⁰. But anything divided by itself is 1, so 5⁰ = 1.

Examples

7⁰ = 1, 1000⁰ = 1, (−3)⁰ = 1, (⅔)⁰ = 1.

Careful

3 × 4⁰ = 3 × 1 = 3, not 1.

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Negative Indices

Number

Lesson 5 of 7

A negative index means "one over": a⁻ⁿ = 1/aⁿ.

Whole numbers

4⁻² = 1/4² = 1/16. 10⁻³ = 1/1000 = 0.001.

It is not negative

2⁻³ = ⅛, a small positive number, not −8.

Fractions flip

(⅔)⁻² = (3/2)² = 9/4. A negative power turns a fraction upside down.

Using the laws

2³ ÷ 2⁵ = 2⁻², and writing it out: (2 × 2 × 2)/(2 × 2 × 2 × 2 × 2) = ¼.

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Negative Powers

Number

Lesson 5 of 7

Work out (a) 4⁻³ (b) (3/5)⁻²

1

(a) Negative power: one over

4⁻³ = 1/4³

2

Work out the power

4³ = 64, so 4⁻³ = 1/64

3

(b) Negative power of a fraction: flip it

(3/5)⁻² = (5/3)²

4

Square the top and the bottom

5²/3² = 25/9

ANSWER

(a) 1/64 (b) 25/9 = 27/9

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PART TWO · HIGHER

Fractional Indices

Powers that are roots. (The printed slides write 8 to the power two-thirds as 8^(2/3).)

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Powers That Are Roots

Number

Lesson 5 of 7

A power of one-half is a square root; a power of one-third is a cube root.

Why

9^(1/2) × 9^(1/2) = 9¹ = 9. The number that multiplies by itself to give 9 is 3, so 9^(1/2) = √9 = 3.

Unit fractions

a^(1/2) = √a, a^(1/3) = ∛a, and a^(1/n) = ⁿ√a.

Examples

25^(1/2) = 5, 64^(1/3) = 4, 81^(1/4) = 3, 32^(1/5) = 2.

Other fractions

a^(m/n) = (ⁿ√a)^m: the bottom is the root, the top is the power. 8^(2/3) = (∛8)² = 2² = 4.

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Three Steps for a Negative Fractional Power

Number

Lesson 5 of 7

Work out 27^(−2/3) one part of the index at a time.

1

Minus: flip

27^(−2/3) = 1/(27^(2/3))

2

Bottom: root

The 3 means cube root: ∛27 = 3

3

Top: power

The 2 means square: 3² = 9

4

Answer

27^(−2/3) = 1/9

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A Fraction to a Fractional Power

Number

Lesson 5 of 7

Work out (16/81)^(−3/4)

1

Minus: flip the fraction

(81/16)^(3/4)

2

Bottom of the index, 4: fourth root of the top and the bottom

∜81 = 3, ∜16 = 2

3

So far

(3/2)³

4

Top of the index, 3: cube

3³/2³ = 27/8

ANSWER

27/8 = 3⅜

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Solving with Powers

Number

Lesson 5 of 7

Find x if 8^x = ¼

1

Write both sides as powers of 2

8 = 2³ and ¼ = 2⁻²

2

So

(2³)^x = 2⁻², that is 2^(3x) = 2⁻²

3

The bases match, so the indices are equal

3x = −2

4

Divide by 3

x = −⅔

ANSWER

x = −⅔

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Match the Power to Its Value

Number

Lesson 5 of 7

Power

Value

5⁰

A

10

1

2⁻³

B

9

2

49^(1/2)

C

1

3

8^(2/3)

D

⅛

4

1000^(1/3)

E

7

5

(⅓)⁻²

F

4

6

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Number

Lesson 5 of 7

CASE STUDY

Carbon Dating and Negative Powers

Living things take in a little radioactive carbon-14 while they are alive. After they die it decays, and every 5730 years - its half-life - half of what is left disappears. After 1 half-life, 2⁻¹ = ½ of the carbon-14 remains; after 2, 2⁻² = ¼; after n half-lives, 2⁻ⁿ. By measuring how much is left, scientists can work out how long ago something died. A sample with ⅛ = 2⁻³ of its carbon-14 left is about 3 × 5730 = 17 190 years old.

5730

Years: the half-life of carbon-14

2⁻³ = ⅛

What is left after 3 half-lives (about 17 190 years)

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Key Terms

Number

Lesson 5 of 7

Zero index

Any non-zero number to the power 0 equals 1.

Negative index

a⁻ⁿ = 1/aⁿ: one over the positive power.

Reciprocal

One divided by a number. The reciprocal of 4 is ¼; of ⅔ is 3/2.

Fractional index (Higher)

A power that is a fraction: the bottom is a root, the top is a power.

nth root

The number that, raised to the power n, gives the original number.

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Number

Lesson 5 of 7

YOUR TASK

Index Sort

10 minutes

Sort these into three groups - "less than 1", "equal to 1" and "more than 1" - then work out each value. 3⁰, 5⁻¹, (½)⁻², 10⁻², (¾)⁰, (2/5)⁻¹. Higher: add 16^(1/2), 8^(−1/3), 27^(2/3) and (¼)^(−1/2).

1

Predict the group first.

2

Then work out the value.

3

Explain one surprise.

WHAT A GOOD ANSWER SHOWS

Less than 1: 5⁻¹ = ⅕, 10⁻² = 1/100, and (Higher) 8^(−1/3) = ½. Equal to 1: 3⁰, (¾)⁰. More than 1: (½)⁻² = 4, (2/5)⁻¹ = 5/2, and (Higher) 16^(1/2) = 4, 27^(2/3) = 9, (¼)^(−1/2) = 2.

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Can I...?

Number

Lesson 5 of 7

Explain why a⁰ = 1.

Work out a negative power of a whole number.

Work out a negative power of a fraction.

Write 1/aⁿ as a negative power.

Work out a^(1/n) as a root. (Higher)

Work out a^(m/n). (Higher)

Work out negative fractional powers. (Higher)

Solve equations such as 8^x = ¼. (Higher)

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Summary & Exam Focus

a⁰ = 1 for any a except 0.

a⁻ⁿ = 1/aⁿ; a negative power of a fraction flips it.

(Higher) a^(1/n) = ⁿ√a.

(Higher) a^(m/n): root with the bottom, power with the top; a minus sign flips.

EXAM FOCUS

Find the value of 125^(−2/3) (2 marks)

Do the three parts in this order - minus (flip), bottom (root), top (power) - and write each step. Taking the root before the power keeps the numbers small.

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