Number and ratio · GCSE Maths
Indices and standard form
Use the index laws with confidence and write large or small numbers in standard form for science-style GCSE questions.
An index counts how many times you multiply the base by itself. Standard form is a number between 1 and 10, times a power of 10.
The important bits
What you need to know
- 1
aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, and (aᵐ)ⁿ = aᵐⁿ. The base must be the same before you add or subtract indices.
- 2
a⁰ = 1 for any non-zero a. a⁻ⁿ = 1/aⁿ. A negative index means reciprocal, not a negative answer.
- 3
a¹/ⁿ is the nth root of a. aᵐ/ⁿ is the nth root of a, then raised to the power m (or the reverse, with care).
- 4
Standard form is a × 10ⁿ where 1 ≤ a < 10 and n is an integer. 45 000 = 4.5 × 10⁴ and 0.0032 = 3.2 × 10⁻³.
- 5
To multiply in standard form, multiply the a values and add the powers of 10, then rewrite if a is no longer between 1 and 10.
- 6
To divide, divide the a values and subtract the powers of 10. To add or subtract, convert so the powers of 10 match, or convert to ordinary numbers if they are small.
- 7
On a calculator, use the ×10ˣ button rather than typing 10^ as if it were a variable. Write the non-calculator method anyway; papers still test it.
- 8
Check the size: 6 × 10⁷ is sixty million, not six million. Count the zeros or the decimal jumps to catch a sign error on n.
Go deeper
Index laws are counting, not decoration
a² × a⁵ means (a × a) × (a × a × a × a × a), which is a⁷. You add the indices because you are counting factors. Division cancels factors, so you subtract. A power of a power, (a³)⁴, is a³ multiplied by itself four times, which is twelve factors, so multiply the indices. These pictures stop students from inventing a² × a⁵ = a¹⁰. Negative indices follow the same counting: a³ ÷ a⁵ = a⁻² = 1/a². The answer is a reciprocal, not minus a². When bases differ, stop. 2³ × 5³ can become (2 × 5)³, but 2³ × 5² cannot be combined into a single power without writing the product out.
Go deeper
Standard form is a place-value costume
Standard form does not change the number; it relocates the decimal point and records the move as a power of 10. Moving the point four places left on 45 000 gives 4.5, so multiply by 10⁴ to restore the value. Moving three places right on 0.0032 gives 3.2, so multiply by 10⁻³. After multiplying 4 × 10⁵ by 6 × 10³ you get 24 × 10⁸, which is not standard form until you write 2.4 × 10⁹. That rewrite is a frequent missed mark. Addition needs matching powers: 3.1 × 10⁴ + 4.2 × 10³ is 3.1 × 10⁴ + 0.42 × 10⁴ = 3.52 × 10⁴. If the powers differ by many places, converting to ordinary numbers can be safer on a non-calculator paper.
See the idea in action
Simplify (3 × 10⁵) × (4 × 10⁻²) and give the answer in standard form. Multiply 3 × 4 = 12 and add the indices: 10⁵ × 10⁻² = 10³. So 12 × 10³ = 1.2 × 10⁴.
Exam technique
Turn knowledge into marks
If your a value is 10 or more, or less than 1, the answer is not yet in standard form. Adjust a and change the power of 10 by the same number of places.
Common mistakes
Do not give these marks away
- 01
Writing a⁻² as −a² instead of 1/a².
- 02
Leaving 12 × 10³ as the final standard-form answer instead of 1.2 × 10⁴.
- 03
Adding indices when multiplying numbers with different bases, such as treating 2³ × 5³ as 10⁹ without combining the bases first.
Which of these is 0.00056 in standard form?
A5.6 × 10⁴
B5.6 × 10⁻⁴
C56 × 10⁻⁵
D5.6 × 10⁻³
Show the answer
5.6 × 10⁻⁴. Move the decimal point four places to the right to get 5.6, so the power is −4. 56 × 10⁻⁵ is equal in value but not standard form, because 56 is not between 1 and 10.
Quick questions
If this is the bit you searched
Why is 10.5 × 10³ not standard form?
The first factor must satisfy 1 ≤ a < 10. Rewrite 10.5 × 10³ as 1.05 × 10⁴.
What does 9¹/² mean?
It is the square root of 9, which is 3 (the principal, positive root in GCSE work unless the question asks for both roots).