FORM 2 • TERM 1 • LESSON 05

Scientists measure things as big as the distance to the Sun and as small as a blood cell. Writing all the zeros is slow and easy to get wrong, so we use powers. In this lesson you will work with numbers in index form, and write very large and very small numbers in standard form, also called scientific notation.

Learning outcomes

By the end of this lesson you should be able to:

• 2.1.2.5 – Multiply, divide, add and subtract numbers written in index form (positive indices).
• 2.1.2.3 – Write numbers in standard form and in scientific notation.
• 2.1.3.7 – Write numbers in scientific notation from a × 10⁻⁶ to a × 10⁶.

Index form

Index form is a short way to write a number multiplied by itself. In 2⁵, the base is 2 and the index (or power) is 5. The index tells you how many 2s are multiplied together:

2⁵ = 2 × 2 × 2 × 2 × 2 = 32.

You read 2⁵ as “2 to the power 5”. Special cases: 7² is “7 squared”, 4³ is “4 cubed”, and any number to the power 1 is itself, so 9¹ = 9.

Multiplying and dividing in index form

When the bases are the same, there are two shortcuts, called the laws of indices:

• Multiply → add the indices: a^m × aⁿ = a^{m + n}. For example 3⁴ × 3² = 3⁶, because (3 × 3 × 3 × 3) × (3 × 3) is six 3s.
• Divide → subtract the indices: a^m ÷ aⁿ = a^{m − n}. For example 5⁶ ÷ 5⁴ = 5² = 25, because four of the 5s cancel.

If the bases are different, such as 2³ × 5², there is no shortcut: work out each power, then multiply: 8 × 25 = 200.

Adding and subtracting in index form

There is no law of indices for adding or subtracting. Work out each power first, then add or subtract:

• 2³ + 2⁴ = 8 + 16 = 24. (It is not 2⁷ = 128.)
• 3⁴ − 3² = 81 − 9 = 72.

Powers of 10

Powers of 10 are the key to standard form. A positive power tells you how many times to multiply by 10; a negative power tells you how many times to divide by 10. For example 10⁻³ = 1 ÷ 10 ÷ 10 ÷ 10 = 0.001.

Power	Ordinary number	In words
106	1,000,000	one million
105	100,000	one hundred thousand
104	10,000	ten thousand
103	1,000	one thousand
102	100	one hundred
101	10	ten
100	1	one
10−1	0.1	one tenth
10−2	0.01	one hundredth
10−3	0.001	one thousandth
10−4	0.0001	one ten-thousandth
10−5	0.00001	one hundred-thousandth
10−6	0.000001	one millionth

Standard form and scientific notation

A number is in standard form when it is written as

a × 10ⁿ, where 1 ≤ a < 10 and n is an integer.

So a must have exactly one non-zero digit before the decimal point. In Caribbean and British books this is called standard form; scientists and calculators call the same thing scientific notation. (Some American books use “standard form” to mean the ordinary way of writing a number, such as 3,400,000, so check which meaning a question is using.)

Large numbers

1. Put the decimal point after the first non-zero digit to get a: 3,400,000 → 3.4.
2. Count how many places the point moved to the left: 6 places.
3. That count is the positive power: 3,400,000 = 3.4 × 10⁶.

Small numbers

1. Put the decimal point after the first non-zero digit: 0.00052 → 5.2.
2. Count how many places the point moved to the right: 4 places.
3. Use a negative power: 0.00052 = 5.2 × 10⁻⁴.

To go back to an ordinary number, move the point the other way: a positive power makes the number bigger (move right) and a negative power makes it smaller (move left). For example 7.05 × 10³ = 7,050 and 6 × 10⁻² = 0.06.

Worked examples

Example 1: Multiplying and dividing powers

Question: Simplify, then work out:
(a) 2³ × 2⁴
(b) 7⁵ ÷ 7³
(c) 10² × 10⁴.

1. (a) Same base, so add the indices: 2⁷ = 128.
2. (b) Subtract the indices: 7² = 49.
3. (c) Add the indices: 10⁶ = 1,000,000.

Answer:
(a) 2⁷ = 128
(b) 7² = 49
(c) 10⁶ = 1,000,000.

Example 2: Adding and subtracting powers

Question: Work out
(a) 5² + 5³
(b) 2⁶ − 4².

1. (a) 25 + 125 = 150.
2. (b) 64 − 16 = 48.

Answer:
(a) 150
(b) 48.

Example 3: A large number in standard form

Question: The population of Trinidad and Tobago is roughly 1,400,000. Write this in standard form.

1. a = 1.4 (one digit before the point).
2. The point moves 6 places to the left.

Answer: 1.4 × 10⁶.

Example 4: A small number in standard form

Question: A human hair is about 0.00008 m thick. Write this in scientific notation.

1. a = 8.
2. The point moves 5 places to the right, so the power is −5.

Answer: 8 × 10⁻⁵ m.

Example 5: Back to ordinary numbers

Question: Write as ordinary numbers:
(a) 2.75 × 10⁴
(b) 9.1 × 10⁻³.

1. (a) Move the point 4 places right: 27,500.
2. (b) Move the point 3 places left: 0.0091.

Answer:
(a) 27,500
(b) 0.0091.

Example 6: Fixing a number that is not in standard form

Question: Write 34 × 10³ correctly in standard form.

1. 34 is not between 1 and 10, so it is not in standard form.
2. 34 = 3.4 × 10, so 34 × 10³ = 3.4 × 10 × 10³ = 3.4 × 10⁴.

Answer: 3.4 × 10⁴ (= 34,000).

Example 7: Comparing numbers in standard form

Question: Which is larger: 4.2 × 10⁵ or 9.8 × 10⁴?

1. Compare the powers first: 10⁵ is bigger than 10⁴.
2. Check: 420,000 and 98,000.

Answer: 4.2 × 10⁵.

Common mistakes

• Multiplying the base and the index. 2⁵ means 2 × 2 × 2 × 2 × 2 = 32, not 2 × 5 = 10.
• Adding indices when you are adding numbers. 2³ + 2⁴ is 24, not 2⁷.
• Using the laws of indices with different bases. 2³ × 3² is not 6⁵; it is 8 × 9 = 72.
• Leaving a outside 1 to 10. 0.52 × 10³ and 52 × 10¹ are not in standard form; the answer is 5.2 × 10².
• Getting the sign of the power wrong. Numbers bigger than 10 have a positive power; numbers between 0 and 1 have a negative power.

Practice questions

Try these yourself first, then open each answer to check your working.

Quick check

Previous: Significant Figures, Estimation and Mental Maths Strategies

Next: Number Bases: Place Value, Expanded Form, Binary and Converting to Base 10

Curriculum link: This post is relevant to the Trinidad and Tobago National Lower Secondary (NCSE) Mathematics Curriculum – Form 2, Term 1, Strand 2.1 Number Operations and Number Theory, Topic 2.1.2 Laws and Properties of Numbers – Learning Outcomes 2.1.2.5 (the four operations with numbers in index form, positive indices) and 2.1.2.3 (standard form and scientific notation); and Topic 2.1.3 Number Bases – Learning Outcome 2.1.3.7 (scientific notation from a × 10⁻⁶ to a × 10⁶).

Attributions to media used in this post:

Red White Blood cells.jpg – Electron Microscopy Facility at The National Cancer Institute at Frederick (NCI-Frederick) – Public domain (via Wikimedia Commons)

Updates

2026/10/23

– First version of this lesson.

2026/10/25

– Put each lettered part of the questions and answers on its own line.

© 2026 Vedesh Kungebeharry. All rights reserved.