Year 3 · Unit 1 · Lesson 2 of 5

Standard Form for Large Numbers: Writing the Distance to the Sun

About 15 minutes · what standard form means · writing large numbers in standard form · back to ordinary numbers · comparing numbers · multiplying and dividing in standard form · no calculator · exit quiz

Hook

The Sun is about 150,000,000 km from the Earth. That is a lot of zeros to write, and easy to miscount.

Is there a shorter way to write 150,000,000 that nobody could misread?

Yes: 1.5 × 10⁸ km. The 1.5 gives the digits, and 10⁸ says the point moves 8 places to the right. This is standard form, the way scientists and calculators write very large (and very small) numbers.

Ordinary numberStandard formPoint moves
3,0003 × 10³3 places
45,0004.5 × 10⁴4 places
150,000,0001.5 × 10⁸8 places

A × 10ⁿ, where A is at least 1 and less than 10.

Learn

Master the essentials

NumberStandard form?
2.3 × 10⁵Yes
23 × 10⁴No: 23 is 10 or more
0.23 × 10⁶No: 0.23 is less than 1
2.3 + 10⁵No: it must be ×

Check A first.

Standard form

A number written as A × 10ⁿ, where A is at least 1 and less than 10.

4.5 × 10⁴ = 45,000

Writing a large number

Put the point after the first digit. Count how many places it moves: that is the power.

7,200 = 7.2 × 10³

Back to an ordinary number

Multiply by 10ⁿ: move the point n places to the right, filling with zeros.

3.06 × 10⁴ = 30,600

Comparing

Compare the powers of 10 first. Only if they are equal, compare the A parts.

9.9 × 10⁴ is less than 1.1 × 10⁵

Multiplying and dividing

Multiply (or divide) the A parts, then use the index laws on the powers of 10.

(2 × 10³) × (3 × 10⁴) = 6 × 10⁷

Tidy the answer

If A ends up 10 or more, divide it by 10 and add 1 to the power.

15 × 10⁶ = 1.5 × 10⁷

Worked examples

See it done

1. Write 150,000,000 in standard form.

  1. Put the point after the first digit: 1.5.
  2. The point moves 8 places to get back to 150,000,000.

1.5 × 10⁸

2. Write 6.02 × 10⁵ as an ordinary number.

  1. Move the point 5 places to the right.
  2. 6.02 → 602,000 (fill the gaps with zeros).

602,000

3. Write 34 × 10⁶ in standard form.

  1. 34 is not between 1 and 10. 34 = 3.4 × 10.
  2. So 34 × 10⁶ = 3.4 × 10¹ × 10⁶ = 3.4 × 10⁷.

3.4 × 10⁷

Common mistake: Forgetting that A must be between 1 and 10. 15 × 10⁷ is the right size but it is not standard form. Make A smaller by ÷ 10 and the power bigger by 1: 1.5 × 10⁸.

PractiseNo calculator

10 questions with instant feedback

Choose the correct answer. Check that A is at least 1 and less than 10. Questions 1–3 are warm-ups, 4–7 are at exam standard and 8–10 are challenge questions. After you answer, the feedback shows the full solution.

Score: 0 / 10Answered: 0 / 10

Warm-up (The basic skill: one or two steps.)

1.Write 7,000 in standard form.

Not quite. 7,000 has three zeros: the point moves 3 places.

Solution

  1. Put the point after the first digit: 7.
  2. The point moves 3 places to make 7,000, so 7 × 10³.

Answer: B. 7 × 10³.

Correct.

Solution

  1. Put the point after the first digit: 7.
  2. The point moves 3 places to make 7,000, so 7 × 10³.

Not quite. 70 is not between 1 and 10.

Solution

  1. Put the point after the first digit: 7.
  2. The point moves 3 places to make 7,000, so 7 × 10³.

Answer: B. 7 × 10³.

Not quite. A must be at least 1.

Solution

  1. Put the point after the first digit: 7.
  2. The point moves 3 places to make 7,000, so 7 × 10³.

Answer: B. 7 × 10³.

2.Write 4.8 × 10⁴ as an ordinary number.

Not quite. Move the point 4 places; don't write 4 zeros after 48.

Solution

  1. Move the point 4 places to the right.
  2. 4.8 → 48 → 480 → 4,800 → 48,000.

Answer: C. 48,000.

Not quite. That is 4.8 × 10³.

Solution

  1. Move the point 4 places to the right.
  2. 4.8 → 48 → 480 → 4,800 → 48,000.

Answer: C. 48,000.

Correct.

Solution

  1. Move the point 4 places to the right.
  2. 4.8 → 48 → 480 → 4,800 → 48,000.

Not quite. 10⁴ is not 4: it means move the point 4 places.

Solution

  1. Move the point 4 places to the right.
  2. 4.8 → 48 → 480 → 4,800 → 48,000.

Answer: C. 48,000.

3.Which number is written in standard form?

Not quite. 36 is not less than 10.

Solution

  1. In standard form, A × 10ⁿ has A at least 1 and less than 10.
  2. 3.6 × 10⁵ is the only one with A between 1 and 10 and a × sign.

Answer: D. 3.6 × 10⁵.

Not quite. 0.36 is less than 1.

Solution

  1. In standard form, A × 10ⁿ has A at least 1 and less than 10.
  2. 3.6 × 10⁵ is the only one with A between 1 and 10 and a × sign.

Answer: D. 3.6 × 10⁵.

Not quite. Standard form multiplies by a power of 10.

Solution

  1. In standard form, A × 10ⁿ has A at least 1 and less than 10.
  2. 3.6 × 10⁵ is the only one with A between 1 and 10 and a × sign.

Answer: D. 3.6 × 10⁵.

Correct.

Solution

  1. In standard form, A × 10ⁿ has A at least 1 and less than 10.
  2. 3.6 × 10⁵ is the only one with A between 1 and 10 and a × sign.

Exam standard (Several steps, like real exam questions.)

4.Write 92,500,000 in standard form.

Not quite. Count again: the point moves 7 places.

Solution

  1. Put the point after the first digit: 9.25.
  2. Count the places back to 92,500,000: 7.
  3. So 9.25 × 10⁷.

Answer: D. 9.25 × 10⁷.

Not quite. 925 is not between 1 and 10.

Solution

  1. Put the point after the first digit: 9.25.
  2. Count the places back to 92,500,000: 7.
  3. So 9.25 × 10⁷.

Answer: D. 9.25 × 10⁷.

Not quite. 92.5 is not less than 10.

Solution

  1. Put the point after the first digit: 9.25.
  2. Count the places back to 92,500,000: 7.
  3. So 9.25 × 10⁷.

Answer: D. 9.25 × 10⁷.

Correct.

Solution

  1. Put the point after the first digit: 9.25.
  2. Count the places back to 92,500,000: 7.
  3. So 9.25 × 10⁷.

5.Write 2.07 × 10⁶ as an ordinary number.

Correct.

Solution

  1. Move the point 6 places to the right.
  2. 2.07 → 2,070,000 (the digits 2, 0, 7, then zeros).

Not quite. That is 2.07 × 10⁷.

Solution

  1. Move the point 6 places to the right.
  2. 2.07 → 2,070,000 (the digits 2, 0, 7, then zeros).

Answer: A. 2,070,000.

Not quite. That is 2.07 × 10⁵.

Solution

  1. Move the point 6 places to the right.
  2. 2.07 → 2,070,000 (the digits 2, 0, 7, then zeros).

Answer: A. 2,070,000.

Not quite. Keep the 0 in 2.07: the digits are 2, 0, 7.

Solution

  1. Move the point 6 places to the right.
  2. 2.07 → 2,070,000 (the digits 2, 0, 7, then zeros).

Answer: A. 2,070,000.

6.Write 58 × 10⁴ in standard form.

Not quite. When A gets 10 times smaller, the power goes up by 1, not down.

Solution

  1. 58 = 5.8 × 10.
  2. So 58 × 10⁴ = 5.8 × 10¹ × 10⁴ = 5.8 × 10⁵.

Answer: B. 5.8 × 10⁵.

Correct.

Solution

  1. 58 = 5.8 × 10.
  2. So 58 × 10⁴ = 5.8 × 10¹ × 10⁴ = 5.8 × 10⁵.

Not quite. 0.58 is less than 1.

Solution

  1. 58 = 5.8 × 10.
  2. So 58 × 10⁴ = 5.8 × 10¹ × 10⁴ = 5.8 × 10⁵.

Answer: B. 5.8 × 10⁵.

Not quite. You made A 10 times smaller, so add 1 to the power.

Solution

  1. 58 = 5.8 × 10.
  2. So 58 × 10⁴ = 5.8 × 10¹ × 10⁴ = 5.8 × 10⁵.

Answer: B. 5.8 × 10⁵.

7.Put these in order, smallest first: 4.1 × 10⁵, 9.8 × 10⁴, 1.2 × 10⁶.

Correct.

Solution

  1. Compare the powers of 10 first: 10⁴ < 10⁵ < 10⁶.
  2. So 9.8 × 10⁴ (98,000) is smallest, then 4.1 × 10⁵ (410,000), then 1.2 × 10⁶ (1,200,000).

Not quite. That is largest first.

Solution

  1. Compare the powers of 10 first: 10⁴ < 10⁵ < 10⁶.
  2. So 9.8 × 10⁴ (98,000) is smallest, then 4.1 × 10⁵ (410,000), then 1.2 × 10⁶ (1,200,000).

Answer: A. 9.8 × 10⁴, 4.1 × 10⁵, 1.2 × 10⁶.

Not quite. Compare the powers first: 10⁴ is smaller than 10⁵.

Solution

  1. Compare the powers of 10 first: 10⁴ < 10⁵ < 10⁶.
  2. So 9.8 × 10⁴ (98,000) is smallest, then 4.1 × 10⁵ (410,000), then 1.2 × 10⁶ (1,200,000).

Answer: A. 9.8 × 10⁴, 4.1 × 10⁵, 1.2 × 10⁶.

Not quite. Look at the powers of 10 first, not the A parts.

Solution

  1. Compare the powers of 10 first: 10⁴ < 10⁵ < 10⁶.
  2. So 9.8 × 10⁴ (98,000) is smallest, then 4.1 × 10⁵ (410,000), then 1.2 × 10⁶ (1,200,000).

Answer: A. 9.8 × 10⁴, 4.1 × 10⁵, 1.2 × 10⁶.

Challenge (Harder problems that mix ideas or need reasoning.)

8.Work out (3 × 10⁴) × (2 × 10⁵). Give your answer in standard form.

Not quite. Add the indices: 4 + 5 = 9.

Solution

  1. Multiply the numbers: 3 × 2 = 6.
  2. Multiply the powers of 10: add the indices, 10⁴ × 10⁵ = 10⁹.
  3. So 6 × 10⁹.

Answer: D. 6 × 10⁹.

Not quite. Multiply 3 × 2 = 6; don't add.

Solution

  1. Multiply the numbers: 3 × 2 = 6.
  2. Multiply the powers of 10: add the indices, 10⁴ × 10⁵ = 10⁹.
  3. So 6 × 10⁹.

Answer: D. 6 × 10⁹.

Not quite. Multiplying: add the indices, don't subtract.

Solution

  1. Multiply the numbers: 3 × 2 = 6.
  2. Multiply the powers of 10: add the indices, 10⁴ × 10⁵ = 10⁹.
  3. So 6 × 10⁹.

Answer: D. 6 × 10⁹.

Correct.

Solution

  1. Multiply the numbers: 3 × 2 = 6.
  2. Multiply the powers of 10: add the indices, 10⁴ × 10⁵ = 10⁹.
  3. So 6 × 10⁹.

9.Work out (8 × 10⁹) ÷ (2 × 10³). Give your answer in standard form.

Not quite. Divide: subtract the indices, 9 − 3 = 6.

Solution

  1. Divide the numbers: 8 ÷ 2 = 4.
  2. Divide the powers of 10: subtract the indices, 10⁹ ÷ 10³ = 10⁶.
  3. So 4 × 10⁶.

Answer: C. 4 × 10⁶.

Not quite. Divide 8 ÷ 2 = 4; don't subtract.

Solution

  1. Divide the numbers: 8 ÷ 2 = 4.
  2. Divide the powers of 10: subtract the indices, 10⁹ ÷ 10³ = 10⁶.
  3. So 4 × 10⁶.

Answer: C. 4 × 10⁶.

Correct.

Solution

  1. Divide the numbers: 8 ÷ 2 = 4.
  2. Divide the powers of 10: subtract the indices, 10⁹ ÷ 10³ = 10⁶.
  3. So 4 × 10⁶.

Not quite. Dividing means subtract the indices, not add.

Solution

  1. Divide the numbers: 8 ÷ 2 = 4.
  2. Divide the powers of 10: subtract the indices, 10⁹ ÷ 10³ = 10⁶.
  3. So 4 × 10⁶.

Answer: C. 4 × 10⁶.

10.Work out (6 × 10⁵) × (4 × 10³). Give your answer in standard form.

Not quite. 24 is not between 1 and 10: tidy it.

Solution

  1. Multiply the numbers: 6 × 4 = 24. Add the indices: 5 + 3 = 8. That gives 24 × 10⁸.
  2. 24 = 2.4 × 10, so add 1 to the power.
  3. The answer is 2.4 × 10⁹.

Answer: C. 2.4 × 10⁹.

Not quite. When 24 becomes 2.4, add 1 to the power.

Solution

  1. Multiply the numbers: 6 × 4 = 24. Add the indices: 5 + 3 = 8. That gives 24 × 10⁸.
  2. 24 = 2.4 × 10, so add 1 to the power.
  3. The answer is 2.4 × 10⁹.

Answer: C. 2.4 × 10⁹.

Correct.

Solution

  1. Multiply the numbers: 6 × 4 = 24. Add the indices: 5 + 3 = 8. That gives 24 × 10⁸.
  2. 24 = 2.4 × 10, so add 1 to the power.
  3. The answer is 2.4 × 10⁹.

Not quite. Add the indices: 5 + 3 = 8.

Solution

  1. Multiply the numbers: 6 × 4 = 24. Add the indices: 5 + 3 = 8. That gives 24 × 10⁸.
  2. 24 = 2.4 × 10, so add 1 to the power.
  3. The answer is 2.4 × 10⁹.

Answer: C. 2.4 × 10⁹.

Apply

Across the Solar System

Mars is about 2.3 × 10⁸ km from the Sun and Jupiter is about 7.8 × 10⁸ km from the Sun. Light travels about 3 × 10⁵ km every second.

1. Write the distance from the Sun to Mars as an ordinary number.

2. How much further from the Sun is Jupiter than Mars? Give your answer in standard form.

3. How many seconds does light from the Sun take to reach Jupiter? Give your answer as an ordinary number.

  1. 230,000,000 km.
  2. 7.8 − 2.3 = 5.5, so 5.5 × 10⁸ km.
  3. (7.8 × 10⁸) ÷ (3 × 10⁵) = 2.6 × 10³ = 2,600 seconds (about 43 minutes).

Next: the Lesson 2 exit quiz. Five questions, recorded on your dashboard. Aim for 4 out of 5.

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