18.5 Calculating with Standard Form

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: multiplying, dividing, adding and subtracting numbers in standard form, with and without a calculator. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Multiply and divide numbers in standard form using the laws of indices
• Adjust an answer so that it is in standard form (1 ≤ A < 10)
• Add and subtract numbers in standard form by changing to ordinary numbers or matching powers
• Use a calculator with standard form and substitute into formulae in real-life problems

Key words

• Standard form: A × 10ⁿ where 1 ≤ A < 10 and n is a whole number.
• Index laws: 10a × 10b = 10a ⁺ b and 10a ÷ 10b = 10a − b.
• Adjust: Rewrite an answer like 24 × 10⁵ as 2.4 × 10⁶ so that A is between 1 and 10.
• Substitute: Replace the letters in a formula with numbers.

Multiplying

Multiply the A numbers, then multiply the powers of 10 by adding the indices. Check that the answer is in standard form.

Worked example: without a calculator

(2 × 10³) × (4 × 10⁵) = (2 × 4) × 10³ ⁺ ⁵ = 8 × 10⁸

Answer: 8 × 10⁸

Worked example: adjusting the answer

(6 × 10⁴) × (3 × 10²) = 18 × 10⁶
18 is not between 1 and 10. 18 = 1.8 × 10, so 18 × 10⁶ = 1.8 × 10⁷.

Answer: 1.8 × 10⁷

Worked example: with a negative power

(5 × 10−³) × (7 × 10⁸) = 35 × 10−³ ⁺ ⁸ = 35 × 10⁵ = 3.5 × 10⁶

Answer: 3.5 × 10⁶

Dividing

Divide the A numbers, then divide the powers of 10 by subtracting the indices.

Worked example: straightforward

(8 × 10⁹) ÷ (2 × 10³) = (8 ÷ 2) × 10⁹ − ³ = 4 × 10⁶

Answer: 4 × 10⁶

Worked example: adjusting a small A

(3 × 10⁵) ÷ (6 × 10²) = 0.5 × 10³
0.5 is less than 1. 0.5 = 5 × 10−¹, so 0.5 × 10³ = 5 × 10².

Answer: 5 × 10²

Adjusting: If A ≥ 10, divide A by 10 and add 1 to the power. If A < 1, multiply A by 10 and take 1 off the power.

Adding and subtracting

You cannot just add the A numbers unless the powers are the same. The safest way: change to ordinary numbers, add or subtract, then change back.

Worked example: adding

(4.2 × 10⁵) + (3 × 10⁴) = 420 000 + 30 000 = 450 000 = 4.5 × 10⁵

Answer: 4.5 × 10⁵

Worked example: subtracting

(6.1 × 10−²) − (8 × 10−³) = 0.061 − 0.008 = 0.053 = 5.3 × 10−²

Answer: 5.3 × 10−²

Using a calculator and formulae

Use the ×10ˣ (or EXP) key to enter standard form. To enter 3.2 × 10⁷ press 3.2, ×10ˣ, 7. Put brackets round each number if you are not sure. Write the final answer properly, not as 3.2E7.

Worked example: light from the Sun

Light travels at 3 × 10⁵ km per second. It takes about 500 seconds to reach Earth. Use distance = speed × time.
distance = (3 × 10⁵) × 500 = (3 × 10⁵) × (5 × 10²) = 15 × 10⁷ = 1.5 × 10⁸ km

Answer: 1.5 × 10⁸ km

Worked example: with a calculator

A country has 6.7 × 10⁷ people and a land area of 2.4 × 10⁵ km². How many people per km²?
(6.7 × 10⁷) ÷ (2.4 × 10⁵) = 279.16…

Answer: about 279 people per km²

Common mistakes

• Multiplying the powers instead of adding them: 10³ × 10⁵ = 10⁸, not 10¹⁵.
• Leaving an answer as 18 × 10⁶ or 0.5 × 10³. Always adjust so 1 ≤ A < 10.
• Adding the A numbers and the powers: 4 × 10⁵ + 3 × 10⁴ is not 7 × 10⁹.
• Typing 3.2 × 10⁷ as 3.2 × 10 ^ 7 without brackets in a division, so the calculator works in the wrong order.

Revision questions

Try these, then check your answers below each one.

Question 1. Work out (3 × 10⁴) × (3 × 10⁵). (Choose from: 9 × 10²⁰, 9 × 10⁹, 6 × 10⁹, 9 × 10¹.)

Answer: 9 × 10⁹

Working: 3 × 3 = 9, 10⁴⁺⁵ = 10⁹. Answer 9 × 10⁹.

Question 2. Liam says, "2 × 10³ + 3 × 10⁴ = 5 × 10⁷." Is Liam correct?

Answer: No

Working: No. 2000 + 30 000 = 32 000 = 3.2 × 10⁴.

Question 3. Work out (4 × 10³) + (2 × 10²). Give your answer as an ordinary number.

Answer: 4200

Working: 4000 + 200 = 4200.

Related lessons

• Previous lesson: 18.4 Writing Small Numbers in Standard Form
• Next lesson: 19.1 Similarity and Enlargement
• In this chapter: 18.1 Multiplying and Dividing Fractions
• In this chapter: 18.2 The Laws of Indices
• In this chapter: 18.3 Writing Large Numbers in Standard Form

Continue in the full course

This lesson is section 18.5 (EGCMF1805) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.