Calculating with Standard Form –

Calculating with standard form extends the basics of writing numbers in standard form to performing arithmetic with them. You may need to multiply, divide, add, or subtract numbers in standard form on both Foundation and Higher GCSE Maths papers.

What Is Calculating with Standard Form?

Once numbers are in standard form (A × 10^n), you can perform calculations by working with the A values and the powers of 10 separately. Multiplication and division are straightforward; addition and subtraction require the powers to match first.

On calculator papers, you can use the EXP or ×10^x button to enter standard form directly. On non-calculator papers, you must show full algebraic working.

Key Formulas

Step-by-Step Method

Multiplying in Standard Form

1. Multiply the A values together.
2. Add the powers of 10.
3. If the new A value is not between 1 and 10, adjust it and update the power.

Dividing in Standard Form

1. Divide the A values.
2. Subtract the second power from the first.
3. Adjust if A falls outside the range 1 ≤ A < 10.

Adding or Subtracting in Standard Form

1. Rewrite both numbers with the same power of 10 (usually the larger power).
2. Add or subtract the A values.
3. Convert the result back to correct standard form if needed.

Worked Example 1 — Foundation Level

Question: Calculate (3 × 10⁴) × (2 × 10³). Give your answer in standard form.

Working:

Step 1 — Multiply A values: 3 × 2 = 6.

Step 2 — Add powers: 4 + 3 = 7.

Step 3 — 6 is between 1 and 10, so no adjustment needed.

Answer: 6 × 10⁷

Worked Example 2 — Higher Level

Question: Calculate (8.4 × 10⁵) ÷ (2.1 × 10⁻²). Give your answer in standard form.

Working:

Step 1 — Divide A values: 8.4 ÷ 2.1 = 4.

Step 2 — Subtract powers: 5 − (−2) = 5 + 2 = 7.

Step 3 — 4 is between 1 and 10, so no adjustment needed.

Answer: 4 × 10⁷

Worked Example 3 — Exam Style

Question: Calculate (4.5 × 10⁶) + (3.2 × 10⁵). Give your answer in standard form.

Working:

Step 1 — Rewrite with the same power. Convert 3.2 × 10⁵ to the power of 10⁶: 3.2 × 10⁵ = 0.32 × 10⁶.

Step 2 — Add: 4.5 × 10⁶ + 0.32 × 10⁶ = 4.82 × 10⁶.

Step 3 — 4.82 is between 1 and 10, so this is already in standard form.

Answer: 4.82 × 10⁶

Common Mistakes

• Adding powers when adding or subtracting numbers. You add powers only when multiplying. For addition/subtraction, you must convert to the same power first.
• Forgetting to adjust when A falls outside the valid range. If multiplying gives 15 × 10⁸, this is not standard form. Write it as 1.5 × 10⁹.
• Subtracting powers in the wrong order when dividing. Always subtract the second power from the first: m − n, not n − m.

Exam Tips

• On calculator papers, use the EXP or ×10^x button — do not type × 10 ^ manually, as this can cause errors.
• Always state your final answer in standard form unless the question asks for an ordinary number.
• Double-check that 1 ≤ A < 10 in your final answer.

Practice Questions

Q1 (Foundation): Calculate (5 × 10³) × (4 × 10²). Give your answer in standard form.

Q2 (Foundation): Calculate (9.6 × 10⁷) ÷ (3.2 × 10⁴). Give your answer in standard form.

Q3 (Higher): Calculate (7.1 × 10⁸) − (4.6 × 10⁷). Give your answer in standard form.

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Related Topics

• Writing in Standard Form
• Standard Form
• Indices and Index Laws

Summary

• To multiply in standard form, multiply the A values and add the powers.
• To divide, divide the A values and subtract the powers.
• To add or subtract, convert both numbers to the same power of 10 first.
• Always adjust the final answer so that 1 ≤ A < 10.
• Use the EXP button on your calculator for standard form entries.