Standard Form Calculating With and Without a Calculator — AQA GCSE Mathematics

SPEC N9 · NUMBER — STANDARD FORM

Standard Form

Calculating With and Without a Calculator

Very large and very small numbers · All four operations · EXP key

What is Standard Form?

STANDARD FORM

A number written as a × 10ⁿ where 1 ≤ a < 10 and n is an integer.

SCIENTIFIC NOTATION

Alternative name for standard form. Same rules apply.

The two rules

Rule 1: The coefficient must be at least 1 but less than 10. Not 0.37, not 14.2 — only 1 ≤ a < 10.

Rule 2: The power must be an integer. Positive for large numbers, negative for small ones.

Large Numbers → Standard Form

Move the decimal point left until only one non-zero digit is to the left of the point. Count the moves — that is the power of 10.

Rule

Large number → move decimal left → positive power of 10.

Number of places moved = the power n.

Small Numbers → Standard Form

Move the decimal point right until one non-zero digit sits to the left. The number of moves is the negative power.

Rule

Small number (<1) → move decimal right → negative power of 10.

The power is negative — not "the number is negative".

⚠️ Nov24 P1 Q2 — modal error: answer of −24. Students confused the negative power with a negative number. 3.71 × 10⁻⁶ is a tiny positive number. The minus sign belongs to the power only.

Standard Form → Ordinary Numbers

Reverse the process: the power tells you how far to move the decimal and which way.

Standard FormDirectionOrdinary Number

4.53 × 10⁴Move decimal right 445 300

6.2 × 10⁻⁴Move decimal left 40.000 62

1.7 × 10²Move decimal right 2170

9.0 × 10⁻²Move decimal left 20.090

Memory aid

Positive power → move right → bigger number (large).

Negative power → move left → smaller number (tiny).

The Coefficient Rule — 1 ≤ a < 10

This is the most frequently tested standard form concept — and the most common place marks are dropped.

1 ≤ a < 10

Answer writtenStandard form?Correct version

35 × 10⁶✗ (35 ≥ 10)3.5 × 10⁷

0.45 × 10⁶✗ (0.45 < 1)4.5 × 10⁵

10 × 10³✗ (10 is not < 10)1.0 × 10⁴

7.2 × 10⁴✓Already correct

⚠️ EXAMINER ALERT — Jun23 P1 Q12(a). "Complete the inequality: ___ ≤ a < ___". Students wrote 0 ≤ a < 10. Lower limit is 1, not 0. The coefficient must be at least 1 — values like 0.45 or 0 are not valid.

Adding & Subtracting

You cannot add standard form numbers with different powers directly. Convert to the same power first.

Method — match powers

Work out (3.2 × 10⁵) + (4.5 × 10⁴)

Rewrite 4.5 × 10⁴ as 0.45 × 10⁵ (same power as the first term).

(3.2 + 0.45) × 10⁵ = 3.65 × 10⁵

Example — Subtraction

Work out (8.1 × 10⁶) − (3.0 × 10⁵)

Rewrite 3.0 × 10⁵ as 0.30 × 10⁶.

(8.1 − 0.30) × 10⁶ = 7.8 × 10⁶

⚠️ EXAMINER ALERT — adding powers instead of numbers. Students write (3 × 10⁴) + (4 × 10⁵) = 7 × 10⁹. That multiplies — it doesn't add. To add: match powers, add coefficients.

Practice Question

Work out (5.4 × 10⁷) + (3.0 × 10⁶).

Give your answer in standard form. 2 marks

M:

Match powers. Rewrite 3.0 × 10⁶ as 0.30 × 10⁷.

C:

(5.4 + 0.30) × 10⁷ = 5.70 × 10⁷

E:

5.7 × 10⁷ ✓ (coefficient 5.7 satisfies 1 ≤ 5.7 < 10)

Multiplying in Standard Form

Multiply coefficients · Add powers

Method

(a × 10ᵐ) × (b × 10ⁿ) = (a × b) × 10ᵐ⁺ⁿ

Example 1

(3 × 10⁴) × (2 × 10⁵)

= (3 × 2) × 10⁴⁺⁵

= 6 × 10⁹ ✓

Example 2 — adjust

(5 × 10³) × (7 × 10⁴)

= 35 × 10⁷

= 3.5 × 10⁸ ✓ (adjust!)

Adjust rule

If the coefficient after multiplying is ≥ 10, divide it by 10 and add 1 to the power. If it is < 1, multiply by 10 and subtract 1 from the power.

Dividing in Standard Form

Divide coefficients · Subtract powers

Method

(a × 10ᵐ) ÷ (b × 10ⁿ) = (a ÷ b) × 10ᵐ⁻ⁿ

Example 1

(8 × 10⁷) ÷ (4 × 10³)

= (8 ÷ 4) × 10⁷⁻³

= 2 × 10⁴ ✓

Example 2 — negatives

(6 × 10⁻²) ÷ (2 × 10³)

= (6 ÷ 2) × 10⁻²⁻³

= 3 × 10⁻⁵ ✓

Negative power arithmetic

−2 − 3 = −5. Subtracting a positive makes the power more negative. Be careful: −2 − (−3) = −2 + 3 = +1.

Adjusting Out-of-Range Results

After any calculation, if the coefficient is ≥ 10 or < 1, adjust before writing the final answer.

Raw resultAdjustmentStandard form

35 × 10⁶35 = 3.5 × 10¹ → add 1 to power3.5 × 10⁷

0.4 × 10⁵0.4 = 4 × 10⁻¹ → subtract 1 from power4 × 10⁴

120 × 10⁻³120 = 1.2 × 10² → add 2 to power1.2 × 10⁻¹

Exam Tip

Always check your final coefficient. If it fails the 1 ≤ a < 10 test, adjust. This step earns or loses the final accuracy mark on almost every standard form question.

MCE Worked Example

The mass of a proton is 1.67 × 10⁻²⁷ kg.

The mass of an electron is 9.11 × 10⁻³¹ kg.

How many times heavier is a proton than an electron? (3 s.f.) 3 marks

M Divide proton mass by electron mass:

(1.67 × 10⁻²⁷) ÷ (9.11 × 10⁻³¹)

C = (1.67 ÷ 9.11) × 10⁻²⁷⁻⁽⁻³¹⁾

= 0.1833… × 10⁴

= 1833.15…

= 1.83315… × 10³

E1830 (3 s.f.)

A proton is approximately 1830 times heavier than an electron.

Power arithmetic note

−27 − (−31) = −27 + 31 = +4. Subtracting a negative adds. Then 0.1833 × 10⁴ must be adjusted: 1.833 × 10³. Round to 3 s.f. at the final step only.

Practice Question

Work out (4.8 × 10⁵) ÷ (1.6 × 10⁻²).

Give your answer in standard form. 2 marks

M:

Divide coefficients; subtract powers:

(4.8 ÷ 1.6) × 10⁵⁻⁽⁻²⁾

C:

Coefficient: 4.8 ÷ 1.6 = 3

Power: 5 − (−2) = 5 + 2 = 7

Result: 3 × 10⁷

E:

3 × 10⁷ ✓ (coefficient 3 satisfies 1 ≤ 3 < 10 — no adjustment needed)

Using a Calculator — EXP Key

On Papers 2 and 3 you can use a calculator. Every GCSE calculator has an EXP or ×10ˣ key.

Entering 3.2 × 10⁵

Press: 3 . 2 EXP 5

The display shows 3.2×10⁵ or 3.2E5.

⚠️ EXAMINER ALERT — calculator misuse.

Students type: 3 . 2 × 1 0 EXP 5

This computes 3.2 × 10 × 10⁵ = 3.2 × 10⁶ — ten times too large.

What to write instead: use the EXP key alone — it already represents "× 10 to the power". Never type "× 10" before EXP.

Reading the display

Your calculator may display the result as 1.2E8 or 1.2×10⁸. In your working, always write it fully: 1.2 × 10⁸. Examiners require the written form, not the calculator's shorthand.

Exam-Style: Jun23 P1 Q12

Part (a): a × 10ⁿ is a number written in standard form.

Complete the inequality for the value of a: ___ ≤ a < ___1 mark

Part (b): b × 10ⁿ represents the number 7200 in standard form.

Work out the value of b × 10⁻ⁿ. Write as an ordinary number. 2 marks

Part (a):

1 ≤ a < 10 B1. Lower limit is 1 (not 0). Upper limit is 10 (strictly less than).

Part (b) — M:

7200 in standard form: b = 7.2, n = 3 (since 7200 = 7.2 × 10³).

Part (b) — C:

b × 10⁻ⁿ = 7.2 × 10⁻³ = 0.0072

Part (b) — E:

0.0072 — write as an ordinary number, not in standard form.

Practice Question

Without a calculator, work out (3 × 10⁴) × (5 × 10⁻⁷).

Give your answer in standard form. 2 marks

M:

Multiply coefficients; add powers:

(3 × 5) × 10⁴⁺⁽⁻⁷⁾

C:

Coefficient: 3 × 5 = 15

Power: 4 + (−7) = −3

Raw result: 15 × 10⁻³

Adjust: 15 = 1.5 × 10¹, so 1.5 × 10⁻³⁺¹ = 1.5 × 10⁻²

E:

1.5 × 10⁻² ✓ Check: 0.015 ordinary number. 3 × 10⁴ = 30 000; 5 × 10⁻⁷ = 0.0000005; product = 0.015 ✓

Common Errors — Don't Lose These Marks

Students often writeCorrect versionError type

0.45 × 10⁶4.5 × 10⁵Coefficient < 1

(3 × 10⁴) + (4 × 10⁵) = 7 × 10⁹= 4.3 × 10⁵Added powers instead of matching them

0.0005 = 5 × 10⁴= 5 × 10⁻⁴Positive power for small number

Lower limit for a is 0Lower limit is 1Coefficient boundary definition

35 × 10⁶ as final answer3.5 × 10⁷Forgot to adjust coefficient

Summary

Key Points — Lesson 1.08

Standard form: a × 10ⁿ where 1 ≤ a < 10 and n is an integer.

Large numbers: move decimal left → positive power. Small: move right → negative power.

Add/subtract: convert to same power, then add/subtract coefficients.

Multiply: multiply coefficients, add powers.

Divide: divide coefficients, subtract powers.

Always adjust if coefficient falls outside 1 ≤ a < 10 after calculation.

Calculator: use EXP key only — never type "× 10 EXP".

Must-know inequality

1 ≤ a < 10

This is the definition. Lower limit = 1. The coefficient is always at least 1 and strictly less than 10.

Power arithmetic

Add for multiply: 10ᵐ × 10ⁿ = 10ᵐ⁺ⁿ

Subtract for divide: 10ᵐ ÷ 10ⁿ = 10ᵐ⁻ⁿ

Watch signs: −27 − (−31) = +4

Exam Tip

After every standard form calculation: read back the coefficient. If it is not between 1 and 10, adjust — one move of the decimal, one change to the power. This step separates full marks from method-only marks.