Calculating with powers indices.pptx

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Calculating with powers (indices)

Number

Lesson 4 of 7

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Last Lesson and Before

Number

Lesson 4 of 7

Answer each one, then check.

1

What is 5²?

2

What is √81?

3

Last lesson: write 72 as a product of prime factors in index form.

4

Last lesson: find the HCF of 12 and 18.

5

Last lesson: find the LCM of 4 and 10.

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Last Lesson and Before - Answers

Number

Lesson 4 of 7

1

What is 5²?

25

2

What is √81?

9

3

Last lesson: write 72 as a product of prime factors in index form.

2³ × 3²

4

Last lesson: find the HCF of 12 and 18.

6

5

Last lesson: find the LCM of 4 and 10.

20

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Learning Objectives

Number

Lesson 4 of 7

1

Know the square numbers to 15² and the cubes of 1, 2, 3, 4, 5 and 10.

2

Find square roots and cube roots, including negative roots.

3

Use index notation for powers.

4

Use the index laws to multiply, divide and raise powers.

5

Use the order of operations with powers and roots, with and without a calculator.

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Powers and Roots

Number

Lesson 4 of 7

A power (or index) tells you how many times a number is multiplied by itself.

Index notation

5³ = 5 × 5 × 5 = 125. The 5 is the base; the 3 is the index or power.

Squares and square roots

7² = 49, so √49 = 7. But the equation x² = 49 has two answers: x = 7 or x = −7, since (−7)² = 49 too.

Cubes and cube roots

4³ = 64, so ∛64 = 4. A cube root of a negative number is negative: ∛(−8) = −2.

Roots undo powers

Squaring and square-rooting are inverse operations, like × and ÷.

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Numbers to Know by Heart

Number

Lesson 4 of 7

Square numbers

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225 (up to 15²).

Cube numbers

1, 8, 27, 64, 125 (1³ to 5³) and 10³ = 1000.

Powers of 2

2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 (2¹ to 2¹⁰).

Powers of 10

10, 100, 1000, 10 000 - the index is the number of zeros.

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Why "Squared" and "Cubed"?

Number

Lesson 4 of 7

3² counts the squares in a square; 3³ counts the cubes in a cube.

A square with sides 3 units long is made of 3 × 3 = 9 unit squares - that is why 3² is called "3 squared". A cube with edges 3 units long is made of 3 × 3 × 3 = 27 unit cubes - "3 cubed".

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PART ONE

The Index Laws

Shortcuts for powers of the same number.

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The Three Index Laws

Number

Lesson 4 of 7

They only work when the base is the same.

Law

In words

Example

Multiplying

Add the indices

3⁴ × 3² = 3⁶

Dividing

Subtract the indices

5⁷ ÷ 5³ = 5⁴

Power of a power

Multiply the indices

(2³)⁴ = 2¹²

Different bases

The laws do not apply: work each power out

2³ × 5² = 8 × 25 = 200

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Why the Laws Work

Number

Lesson 4 of 7

Write the powers out in full and count.

Multiplying

3⁴ × 3² = (3 × 3 × 3 × 3) × (3 × 3): six 3s multiplied, so 3⁶.

Dividing

5⁷ ÷ 5³: three of the seven 5s cancel, leaving four: 5⁴.

Power of a power

(2³)⁴ = 2³ × 2³ × 2³ × 2³: four lots of three 2s, so 2¹².

In letters

a^m × aⁿ = a^(m+n), a^m ÷ aⁿ = a^(m−n) and (a^m)ⁿ = a^(mn).

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Using the Index Laws

Number

Lesson 4 of 7

Work out the value of (7⁵ × 7³)/7⁶

1

Multiplying: add the indices

7⁵ × 7³ = 7⁸

2

Dividing: subtract the indices

7⁸ ÷ 7⁶ = 7²

3

Work out the value

7² = 49

ANSWER

49

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Changing the Base

Number

Lesson 4 of 7

Write 4³ × 8² as a single power of 2.

1

Write each base as a power of 2

4 = 2² and 8 = 2³

2

Power of a power: multiply

4³ = (2²)³ = 2⁶ and 8² = (2³)² = 2⁶

3

Multiplying: add

2⁶ × 2⁶ = 2¹²

4

Check

4³ × 8² = 64 × 64 = 4096 = 2¹²

ANSWER

2¹²

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PART TWO

Calculating with Powers

Order of operations, with and without a calculator.

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Order of Operations

Number

Lesson 4 of 7

Powers and roots come before multiplying, dividing, adding and subtracting.

BIDMAS

Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.

Negative numbers

−3² = −9 (square 3, then make it negative), but (−3)² = 9.

A root sign is a bracket

√(9 + 16) = √25 = 5. It is NOT √9 + √16 = 7.

On a calculator

Use brackets for the top and bottom of a fraction, and write down the full display before rounding.

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A Non-Calculator Calculation

Number

Lesson 4 of 7

Work out 2³ × 5² − √144

1

Powers and roots first

2³ = 8, 5² = 25, √144 = 12

2

Then multiply

8 × 25 = 200

3

Then subtract

200 − 12 = 188

ANSWER

188

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Mistakes to Avoid

Number

Lesson 4 of 7

WRONG

• 3⁴ × 3² = 9⁶

• 2³ = 6

• 5² + 5³ = 5⁵

• (−4)² = −16

RIGHT

• 3⁴ × 3² = 3⁶ - the base stays the same.

• 2³ = 2 × 2 × 2 = 8

• 5² + 5³ = 25 + 125 = 150 - no law for adding.

• (−4)² = 16 - negative times negative is positive.

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Number

Lesson 4 of 7

CASE STUDY

Rice on a Chessboard

An old legend tells of a clever inventor who showed a king the game of chess. As a reward he asked for one grain of rice on the first square of the board, two on the second, four on the third, and so on, doubling each time. The king laughed at such a small request - until his treasurers did the maths. The 64th square alone needs 2⁶³ grains, and the whole board needs 2⁶⁴ − 1: about 18 quintillion grains, far more rice than has ever been grown. The story is a legend, but the numbers are real, and they show how fast powers grow.

2⁶³

Grains on the last square alone

1.8 × 10¹⁹

Grains on the whole board

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Key Terms

Number

Lesson 4 of 7

Power (index)

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

Base

The number being multiplied, e.g. the 5 in 5³.

Square number

A number made by multiplying a whole number by itself, e.g. 49 = 7².

Cube number

A number made by multiplying a whole number by itself three times, e.g. 64 = 4³.

Square root

The number that squares to give a number: √49 = 7.

Cube root

The number that cubes to give a number: ∛64 = 4.

Index laws

The rules for multiplying, dividing and raising powers of the same base.

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Number

Lesson 4 of 7

YOUR TASK

Power Pyramids

10 minutes

Write each as a single power, then as an ordinary number. (a) 2⁵ × 2³ (b) 10⁸ ÷ 10⁵ (c) (3²)³ (d) 5⁹ ÷ (5² × 5⁴) (e) 9² × 3³ as a power of 3 (f) find n if 2ⁿ = 4⁵ ÷ 2³

1

Use the index laws.

2

Change a base where you need to.

3

Check one answer the long way.

WHAT A GOOD ANSWER SHOWS

(a) 2⁸ = 256 (b) 10³ = 1000 (c) 3⁶ = 729 (d) 5³ = 125 (e) 3⁴ × 3³ = 3⁷ = 2187 (f) 4⁵ = 2¹⁰, so 2¹⁰ ÷ 2³ = 2⁷ and n = 7

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Can I...?

Number

Lesson 4 of 7

Recall square numbers to 15².

Recall the cubes of 1, 2, 3, 4, 5 and 10.

Find square roots and cube roots.

Give both square roots of a number.

Multiply powers of the same base.

Divide powers of the same base.

Raise a power to a power.

Use BIDMAS with powers and roots.

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Summary & Exam Focus

5³ means 5 × 5 × 5; a root undoes a power.

Same base: multiply, add indices; divide, subtract indices; power of a power, multiply indices.

Different bases: work each power out separately.

Powers and roots come before ×, ÷, + and −; a root sign acts as a bracket.

EXAM FOCUS

Show that 9⁴ × 27² = 3¹⁴ (3 marks)

In a "show that" question the answer is given, so the marks are all for the working. Write every step, including 9 = 3² and 27 = 3³.

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