Powers of 10 and standard form - Teacher Notes.docx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Powers of 10 and standard form

Number · Lesson 6 of 7

Teacher copy - includes the notes for whoever is teaching from it.

Last Lesson and Before

Answer each one, then check.

1. What is 10³?

1000

2. Last lesson: what is 10⁻²?

1/100 = 0.01

3. Work out 4.5 × 100.

450

4. Write 10⁵ × 10³ as a single power of 10.

10⁸

5. Last lesson: what is 5⁰?

1

Learning Objectives

1. Write positive and negative powers of 10 as ordinary numbers.

2. Use metric prefixes such as kilo, mega, milli and micro.

3. Write numbers in standard form, and change them back.

4. Multiply and divide numbers in standard form.

5. Add and subtract numbers in standard form, and use a calculator with them.

Powers of 10

Multiplying by 10ⁿ moves every digit n places.

▸ Positive powers. 10⁶ = 1 000 000. 3.2 × 10⁴ = 32 000: the digits move 4 places left.

▸ Negative powers. 10⁻³ = 1/1000 = 0.001. 3.2 × 10⁻⁴ = 0.000 32: the digits move 4 places right.

▸ Multiplying powers of 10. 10⁵ × 10³ = 10⁸. Add the indices, as for any base.

▸ Dividing powers of 10. 10⁵ ÷ 10⁸ = 10⁻³. Subtract the indices.

Metric Prefixes

Each prefix stands for a power of 10.

Prefix	Symbol	Power of 10	Example
giga	G	10⁹	1 gigabyte = 1 000 000 000 bytes
mega	M	10⁶	1 megawatt = 1 000 000 watts
kilo	k	10³	1 kilometre = 1000 metres
milli	m	10⁻³	1 millimetre = 0.001 metres
micro	μ	10⁻⁶	1 micrometre = 0.000 001 metres
nano	n	10⁻⁹	1 nanosecond = 0.000 000 001 seconds

What Is Standard Form?

A number in standard form is written as A × 10ⁿ, where 1 ≤ A < 10 and n is a whole number.

▸ The first part. A number from 1 up to (but not including) 10: one non-zero digit before the decimal point.

▸ The power. How many places the digits have moved. Positive for big numbers, negative for small numbers (less than 1).

▸ Big numbers. 45 000 = 4.5 × 10⁴.

▸ Small numbers. 0.0072 = 7.2 × 10⁻³.

Writing Numbers in Standard Form

Number	Standard form	How
45 000	4.5 × 10⁴	4.5 moved 4 places left
3 080 000	3.08 × 10⁶	3.08 moved 6 places left
0.0072	7.2 × 10⁻³	7.2 moved 3 places right
0.000 050 1	5.01 × 10⁻⁵	5.01 moved 5 places right
38 × 10⁴	3.8 × 10⁵	38 = 3.8 × 10¹, so add 1 to the power
0.6 × 10⁻²	6 × 10⁻³	0.6 = 6 × 10⁻¹, so subtract 1 from the power

Is It in Standard Form?

YES	NO - AND WHY
▸ 3.2 × 10⁵ ▸ 9 × 10⁻⁴ ▸ 1.07 × 10¹² ▸ 5 × 10⁰, which is 5	▸ 32 × 10⁴: 32 is not less than 10. ▸ 0.9 × 10⁻³: 0.9 is less than 1. ▸ 1.07 × 5¹²: it must be a power of 10. ▸ 4.5 × 10^(2.5): the power must be a whole number.

From Tiny to Huge

Standard form lets the whole universe fit on one scale.

10⁻¹⁰ M ——▸ 10²¹ M

1. An atom. About 1 × 10⁻¹⁰ m across

2. A red blood cell. About 8 × 10⁻⁶ m across

3. A person. About 1.7 × 10⁰ m (1.7 m) tall

4. The Earth. About 1.3 × 10⁷ m across

5. Our galaxy. About 1 × 10²¹ m across

PART TWO

Calculating in Standard Form

Deal with the numbers and the powers separately.

Multiplying in Standard Form

Work out (3 × 10⁵) × (6 × 10⁻²). Give your answer in standard form.

 

1. Multiply the numbers

3 × 6 = 18

2. Multiply the powers: add the indices

10⁵ × 10⁻² = 10³

3. Put them together

18 × 10³

4. 18 is not between 1 and 10: 18 = 1.8 × 10¹

1.8 × 10¹ × 10³

Answer: 1.8 × 10⁴

Dividing in Standard Form

Work out (4.8 × 10⁶) ÷ (1.2 × 10⁻³). Give your answer in standard form.

 

1. Divide the numbers

4.8 ÷ 1.2 = 4

2. Divide the powers: subtract the indices

10⁶ ÷ 10⁻³ = 10⁶⁻⁽⁻³⁾ = 10⁹

3. Put them together

4 × 10⁹

Answer: 4 × 10⁹

Adding in Standard Form

Work out (4.5 × 10⁴) + (3 × 10³). Give your answer in standard form.

 

1. The powers are different, so write both as ordinary numbers

45 000 and 3000

2. Add

45 000 + 3000 = 48 000

3. Write the answer in standard form

4.8 × 10⁴

Answer: 4.8 × 10⁴

Standard Form on a Calculator

Calculators have a key for entering powers of 10.

▸ The key. Look for the "×10 to the x" key (EXP on some models). To enter 3.2 × 10⁻⁵, press 3.2, then that key, then −5.

▸ Reading the display. A display like 1.8E4 means 1.8 × 10⁴. Always write the answer with × 10 and a power.

▸ Brackets. When dividing by a number in standard form, put it in brackets, or use the power-of-10 key.

▸ Check. Estimate first: is the power of 10 about right?

Case Study

CASE STUDY How Long Does Sunlight Take to Reach Us? The Sun is about 1.5 × 10⁸ km from the Earth, and light travels at about 3 × 10⁵ km per second. Time = distance ÷ speed = (1.5 × 10⁸) ÷ (3 × 10⁵) = 0.5 × 10³ = 5 × 10² seconds: about 500 seconds, or 8 minutes 20 seconds. So when you look at the Sun, you are seeing it as it was more than 8 minutes ago. Standard form turns a calculation with 17 digits into two small divisions.

 

1.5 × 10⁸ km Distance from the Earth to the Sun

5 × 10² s Time for sunlight to reach us - about 8 minutes

Key Terms

Standard form A number written as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer.	Power of 10 10 multiplied by itself a number of times, e.g. 10³ = 1000.
Prefix Letters in front of a unit that stand for a power of 10, e.g. kilo = 10³.	Ordinary number A number written out in full, without powers of 10.
Integer A whole number, positive, negative or zero.

Your Task: Space and Cells

12 minutes

Use standard form for each question, without a calculator. (a) Write 6 000 000 000 000 in standard form. (b) A bacterium is 2 × 10⁻⁶ m long. How many would fit end to end along 1 cm (1 × 10⁻² m)? (c) Put in order, smallest first: 3.1 × 10⁴, 2.9 × 10⁵, 31 000, 0.3 × 10⁵. (d) Work out (2.5 × 10³) × (4 × 10⁶). 1. Write each number in standard form first. 2. Deal with the numbers and the powers separately. 3. Check the first part is between 1 and 10.

A good answer shows: (a) 6 × 10¹². (b) (1 × 10⁻²) ÷ (2 × 10⁻⁶) = 0.5 × 10⁴ = 5 × 10³ = 5000. (c) 0.3 × 10⁵ (30 000), then 3.1 × 10⁴ and 31 000 (equal), then 2.9 × 10⁵. (d) 10 × 10⁹ = 1 × 10¹⁰.

Note: Part (c) has two equal values - a check that students convert properly rather than comparing the first digits.

Can I...?

☐ Write powers of 10 as ordinary numbers.

☐ Use metric prefixes.

☐ Write a large number in standard form.

☐ Write a small number in standard form.

☐ Change standard form back to an ordinary number.

☐ Multiply and divide in standard form.

☐ Add and subtract in standard form.

☐ Use the power-of-10 key on a calculator.

Summary

✓ Standard form: A × 10ⁿ with 1 ≤ A < 10.

✓ Positive powers for big numbers, negative powers for numbers less than 1.

✓ Multiply or divide the numbers and the powers separately, then adjust.

✓ To add or subtract, make the powers match or use ordinary numbers.

 

EXAM FOCUS Work out (5 × 10⁻³) × (7 × 10⁸). Give your answer in standard form. (2 marks) After multiplying or dividing, check the first number is between 1 and 10. 35 × 10⁵ is a correct value but not standard form, and loses the accuracy mark.

Exam Practice: Standard Form

Answer all questions. Show your working. Questions 1 to 5 are non-calculator. · 25 minutes

▸ Question 1 · 1 mark · Non-calculator. Write 0.000 604 in standard form.

▸ Question 2 · 1 mark · Non-calculator. Write 7.3 × 10⁵ as an ordinary number.

▸ Question 3 · 2 marks · Non-calculator. Work out (5 × 10⁻³) × (7 × 10⁸). Give your answer in standard form.

▸ Question 4 · 2 marks · Non-calculator. Work out (8.4 × 10⁷) ÷ (2.1 × 10⁻²). Give your answer in standard form.

▸ Question 5 · 2 marks · Non-calculator. Write these numbers in order of size, starting with the smallest. 4.1 × 10⁻³, 0.0039, 42 × 10⁻⁴, 0.4 × 10⁻²

▸ Question 6 · 3 marks · Calculator. The mass of one atom of gold is 3.27 × 10⁻²² grams. How many atoms of gold are there in 1 kilogram of gold? Give your answer in standard…

Question 1 · 1 mark · Non-calculator

“Write 0.000 604 in standard form.”

HOW TO ANSWER IT Command word: Non-calculator. Worth 1 mark, so plan before writing.

Question 1 · mark scheme

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

▸ 6.04 × 10⁻⁴. B1

▸ Model answer. 6.04 × 10⁻⁴

Question 2 · 1 mark · Non-calculator

“Write 7.3 × 10⁵ as an ordinary number.”

HOW TO ANSWER IT Command word: Non-calculator. Worth 1 mark, so plan before writing.

Question 2 · mark scheme

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

▸ 730 000. B1

▸ Model answer. 730 000

Question 3 · 2 marks · Non-calculator

“Work out (5 × 10⁻³) × (7 × 10⁸). Give your answer in standard form.”

HOW TO ANSWER IT Command word: Non-calculator. Worth 2 marks, so plan before writing.

Question 3 · mark scheme

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

▸ 35 × 10⁵ or 3 500 000 seen. M1

▸ 3.5 × 10⁶. A1

▸ Model answer. 5 × 7 = 35 and 10⁻³ × 10⁸ = 10⁵, so 35 × 10⁵ = 3.5 × 10⁶.

Question 4 · 2 marks · Non-calculator

“Work out (8.4 × 10⁷) ÷ (2.1 × 10⁻²). Give your answer in standard form.”

HOW TO ANSWER IT Command word: Non-calculator. Worth 2 marks, so plan before writing.

Question 4 · mark scheme

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

▸ 4 seen, or 10⁹ seen. M1

▸ 4 × 10⁹. A1

▸ Model answer. 8.4 ÷ 2.1 = 4 and 10⁷ ÷ 10⁻² = 10⁹, so 4 × 10⁹.

Question 5 · 2 marks · Non-calculator

“Write these numbers in order of size, starting with the smallest. 4.1 × 10⁻³, 0.0039, 42 × 10⁻⁴, 0.4 × 10⁻²”

HOW TO ANSWER IT Command word: Non-calculator. Worth 2 marks, so plan before writing.

Question 5 · mark scheme

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

▸ All four in the correct order. B2

▸ At least three converted correctly to the same form, or one pair out of order. B1

▸ Model answer. As ordinary numbers: 0.0041, 0.0039, 0.0042, 0.004. Order: 0.0039, 0.4 × 10⁻², 4.1 × 10⁻³, 42 × 10⁻⁴.

Question 6 · 3 marks · Calculator

“The mass of one atom of gold is 3.27 × 10⁻²² grams. How many atoms of gold are there in 1 kilogram of gold? Give your answer in standard form, correct to 3 significant figures.”

HOW TO ANSWER IT Command word: Calculator. Worth 3 marks, so plan before writing.

Question 6 · mark scheme

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

▸ 1 kg = 1000 g used. P1

▸ 1000 ÷ (3.27 × 10⁻²²). P1

▸ 3.06 × 10²⁴. A1

▸ Model answer. 1 kg = 1000 g. 1000 ÷ (3.27 × 10⁻²²) = 3.058… × 10²⁴, so about 3.06 × 10²⁴ atoms.