Standard Form Calculations — GCSE Maths

A number in standard form is written as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer. Once numbers are in this form, you calculate with them by applying index laws to the powers of 10 and ordinary arithmetic to the A values — keeping the answer in standard form throughout.

What is standard form and why does it matter for calculations?

Standard form (also called standard index form or scientific notation) expresses very large or very small numbers compactly. The key property that makes calculations work is that powers of 10 obey index laws:

• 10ᵃ × 10ᵇ = 10^(a + b)
• 10ᵃ ÷ 10ᵇ = 10^(a − b)

This means you can handle the powers and the A values separately, then recombine.

How do you multiply numbers in standard form?

Step 1 — Multiply the A values.
Step 2 — Add the powers of 10.
Step 3 — Adjust if the A value is not between 1 and 10.

Worked example 1: calculate (3 × 10⁴) × (2 × 10⁵)

Step 1 — Multiply A values: 3 × 2 = 6.
Step 2 — Add powers: 10⁴ × 10⁵ = 10⁹.
Step 3 — 6 is already between 1 and 10, no adjustment needed.
Answer: 6 × 10⁹

Worked example 2: calculate (4 × 10⁶) × (5 × 10³)

Step 1 — 4 × 5 = 20.
Step 2 — 10⁶ × 10³ = 10⁹.
Step 3 — 20 is not between 1 and 10. Write 20 = 2 × 10¹, so:
20 × 10⁹ = 2 × 10¹ × 10⁹ = 2 × 10¹⁰.
Answer: 2 × 10¹⁰

The adjustment step is the most common source of errors — always check that the A value sits between 1 and 10.

How do you divide numbers in standard form?

Step 1 — Divide the A values.
Step 2 — Subtract the powers of 10.
Step 3 — Adjust if necessary.

Worked example 3: calculate (9 × 10⁸) ÷ (3 × 10⁵)

Step 1 — 9 ÷ 3 = 3.
Step 2 — 10⁸ ÷ 10⁵ = 10^(8 − 5) = 10³.
Answer: 3 × 10³

Worked example 4: calculate (2.4 × 10⁷) ÷ (6 × 10⁴)

Step 1 — 2.4 ÷ 6 = 0.4.
Step 2 — 10⁷ ÷ 10⁴ = 10³.
Step 3 — 0.4 is not ≥ 1. Write 0.4 = 4 × 10⁻¹, so:
0.4 × 10³ = 4 × 10⁻¹ × 10³ = 4 × 10².
Answer: 4 × 10²

How do you add or subtract numbers in standard form?

Unlike multiplication and division, adding and subtracting require the powers of 10 to be equal before you combine the A values.

Step 1 — Identify the larger power.
Step 2 — Rewrite the smaller number so that its power of 10 matches the larger.
Step 3 — Add or subtract the A values.
Step 4 — Write in correct standard form.

Worked example 5: calculate (5 × 10⁶) + (3 × 10⁵)

Step 1 — Larger power is 10⁶.
Step 2 — Rewrite 3 × 10⁵ = 0.3 × 10⁶.
Step 3 — (5 + 0.3) × 10⁶ = 5.3 × 10⁶.
Answer: 5.3 × 10⁶

Worked example 6: calculate (8.2 × 10⁴) − (4 × 10³)

Rewrite 4 × 10³ = 0.4 × 10⁴.
(8.2 − 0.4) × 10⁴ = 7.8 × 10⁴.
Answer: 7.8 × 10⁴

Summary of standard form calculation rules

How do you use a calculator with standard form?

On a Casio scientific calculator, enter standard form using the ×10ˣ (or EXP) key:

• To enter 3.6 × 10⁸: press 3 . 6 then ×10ˣ then 8.
• The display shows 3.6E8 or 3.6 ×10⁸ depending on the model.
• Compute normally; if the answer displays as 3.456E11, write it as 3.456 × 10¹¹.

Do not type 3.6 × 10 ^ 8 as a multiplication — pressing ×10ˣ does this in one step and avoids accidental bracket errors.

Frequently asked questions

What if my answer has a negative power of 10?

A negative index represents a very small number: 10⁻³ = 0.001. The rules for calculating are identical — just be careful with signs when adding or subtracting indices. For example, 10⁻² × 10⁻⁵ = 10^(−2 + (−5)) = 10⁻⁷.

Why can't I just add two standard form numbers directly?

Because 3 × 10⁴ + 2 × 10³ ≠ 5 × 10⁷. The powers are separate multipliers, not values you add — just as you would not add 3 apples and 2 oranges and call the total 5 apples. You must align the place values (powers) before combining the counts (A values).

How do I know when the adjustment step is needed?

After multiplying or dividing the A values, check: is the result between 1 and 10? If yes, no adjustment. If the result is ≥ 10, divide it by 10 and increase the power by 1. If the result is less than 1, multiply it by 10 and decrease the power by 1. Repeat until 1 ≤ A < 10.

Are standard form calculations in the non-calculator paper?

Yes. Multiplication and division appear in the non-calculator paper — which is why the manual method (separate the A and power components, apply index laws, adjust) is essential. Addition and subtraction also appear without a calculator and require the power-alignment step. The calculator paper may have larger or more complex numbers but the method is the same.

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