Place Value, Powers of 10 & Rounding

Cambridge Lower Secondary Stage 7 — Unit 3. Master how digits shift, how to write huge and tiny numbers neatly, and how to round intelligently.

Place Value Table

Watch what happens when you multiply or divide by a power of 10. Each button slides the digits left (multiply) or right (divide).

Millions
10&sup6;		 Hundred Thousands
10&sup5;		 Ten Thousands
10&sup4;		 Thousands
10³		 Hundreds
10²		 Tens
10¹		 Units
10&sup0;		 . Tenths
10⁻¹		 Hundredths
10⁻²		 Thousandths
10⁻³
3 . 4 5

Currently showing: 3.45

The Golden Rules

  • Multiplying by 10n: Digits shift n places to the LEFT (the number gets bigger). The decimal point stays put; it's the digits that move!
  • Dividing by 10n: Digits shift n places to the RIGHT (the number gets smaller).
  • Negative powers: 10-1 = 0.1 = 1/10, 10-2 = 0.01 = 1/100, 10-3 = 0.001 = 1/1000

Standard Form (Scientific Notation)

Any number can be written as: a × 10n

  • a must satisfy: 1 ≤ a < 10 (one non-zero digit before the decimal point)
  • n is any integer (positive, negative, or zero)
  • Large numbers: n is positive.   Small numbers (less than 1): n is negative.

Example:   4 500 000 = 4.5 × 106    |    0.0034 = 3.4 × 10-3

Rounding to Significant Figures

  • 1st sig fig = first non-zero digit from the left.
  • Count the required number of sig figs from there.
  • Look at the next digit: if it is 5 or more, round up; otherwise, round down (leave unchanged).
  • Replace any remaining digits before the decimal point with zeros as placeholders.

47 382 rounded to 2 s.f. = 47 000  |  0.003 671 to 2 s.f. = 0.0037

Worked Examples

Study each worked solution carefully before you practise.

Example 1: Multiply by 103

Question: Calculate 6.72 × 103.

Method: 103 = 1 000. Multiplying by 1 000 moves digits 3 places left.

6.72 → 67.2 (× 10) → 672 (× 100) → 6 720 (× 1 000)

Answer: 6 720

Example 2: Divide by 102 (multiply by 10-2)

Question: Calculate 58.3 ÷ 102. (Equivalent to 58.3 × 10-2.)

Method: Dividing by 100 moves digits 2 places right.

58.3 → 5.83 (÷ 10) → 0.583 (÷ 100)

Answer: 0.583

Example 3: Write in Standard Form

Question: Write 0.000 047 in standard form.

Method:

  1. Find the first non-zero digit: 4.
  2. Write a: place the decimal point after the first non-zero digit → 4.7.
  3. Count how many places the decimal moved: from 0.000047 to 4.7 is 5 places right, so the power is -5.

Answer: 4.7 × 10-5

Example 4: Round to 2 Significant Figures

Question: Round 0.083 461 to 2 significant figures.

Method:

  1. First sig fig = 8 (first non-zero digit).
  2. Second sig fig = 3.
  3. Look at the next digit: 4 < 5, so round down (leave 3 unchanged).

Answer: 0.083

Standard Form Machine

Type any ordinary number and convert it to standard form — or type a standard form expression and expand it. Watch the machine work!

Type an ordinary number (e.g. 0.00056 or 3400000):

Result appears here...

Or type standard form (enter a and n) to expand:

× 10n
Expanded number appears here...

Ex 1: Powers of 10 Calculator

Type your answer for each calculation. Click Check when ready!

Ex 2: Write in Standard Form

Write each ordinary number in the form a × 10n. Enter a (between 1 and 10) and n (the integer power) separately.

Ex 3: Expand Standard Form

Convert each standard form number back to an ordinary number. Write exactly (e.g. 0.0034, not 3.4e-3).

Ex 4: Rounding to Significant Figures

Round each number to the stated number of significant figures. Immediate feedback given!

Ex 5: Order of Magnitude

Click the numbers in order from SMALLEST to LARGEST. Each click places the next number in the ranked list. Click Reset to try again!

Click cards in order: smallest first → largest last

Your ranking (smallest → largest):

Ex 6: True or False?

Click TRUE or FALSE for each statement. Immediate feedback!

Practice Questions

Complete all 20 questions on paper. Show full workings!

  1. Calculate 3.7 × 104.
  2. Calculate 0.056 × 103.
  3. Calculate 850 ÷ 102.
  4. Calculate 4.2 × 10-2.
  5. Calculate 7.8 ÷ 10-1.
  6. Write 67 400 in standard form.
  7. Write 0.000 93 in standard form.
  8. Write 8.05 × 104 as an ordinary number.
  9. Write 2.3 × 10-5 as an ordinary number.
  10. Round 3 847 to 1 significant figure.
  11. Round 0.007 361 to 2 significant figures.
  12. Round 45.68 to 3 significant figures.
  13. Round 0.090 54 to 1 significant figure.
  14. A scientist measures a cell as 0.000 023 m. Write this in standard form.
  15. Write 1.06 × 103 as an ordinary number.
  16. Arrange these in ascending order: 3.2 × 104, 4.1 × 103, 2.9 × 105, 8 × 102.
  17. Calculate (5.4 × 103) × (2 × 102). Give your answer in standard form.
  18. Round 0.004 987 to 3 significant figures.
  19. A factory produces 1 200 000 bolts per year. Write this in standard form.
  20. True or false: 0.35 written in standard form is 3.5 × 10-1. Explain your answer.
  1. 37 000
  2. 56
  3. 8.5
  4. 0.042
  5. 78  (dividing by 10-1 = multiplying by 10)
  6. 6.74 × 104
  7. 9.3 × 10-4
  8. 80 500
  9. 0.000 023
  10. 4 000
  11. 0.0074
  12. 45.7
  13. 0.09
  14. 2.3 × 10-5 m
  15. 1 060
  16. 8 × 102 < 4.1 × 103 < 3.2 × 104 < 2.9 × 105
  17. 5.4 × 2 = 10.8; 103 × 102 = 105; 10.8 × 105 = 1.08 × 106
  18. 0.00499
  19. 1.2 × 106
  20. True. 0.35 = 3.5 × 10-1. The value of a is 3.5 (between 1 and 10) and n = -1 because the decimal shifted one place right.

Challenge: Word Problems

Apply your knowledge to real-world science contexts. Show all workings on paper!

  1. The speed of light is approximately 300 000 000 m/s. Write this in standard form.
  2. An atom of hydrogen has a radius of about 0.000 000 000 053 m. Write this in standard form.
  3. The distance from Earth to the Sun is about 1.5 × 1011 m. Write this as an ordinary number.
  4. A red blood cell is approximately 7 × 10-6 m in diameter. How many red blood cells placed end-to-end would fit in 0.014 m? Give your answer in standard form.
  5. The mass of the Earth is 5.97 × 1024 kg. The mass of the Moon is 7.35 × 1022 kg. How many times heavier is the Earth than the Moon? Give your answer to 3 significant figures.
  6. A computer chip performs 2.8 × 109 operations per second. How many operations does it perform in one hour? Give your answer in standard form.
  7. A grain of sand has a mass of about 6.7 × 10-5 g. A beach contains approximately 7.5 × 1018 grains of sand. Estimate the total mass of the sand in standard form (to 2 s.f.).
  8. Light travels at 3 × 108 m/s. The nearest star (Proxima Centauri) is 4.01 × 1016 m away. How many seconds does it take light to travel this distance? Give your answer in standard form to 3 s.f.
  9. A nanoparticle has a diameter of 5 × 10-8 m. A virus has a diameter of 1.5 × 10-7 m. How many times wider is the virus than the nanoparticle?
  10. The world population is approximately 8 × 109. Each person uses roughly 1 250 litres of water per day. Estimate the total daily water usage of the world population in standard form (to 2 s.f.).
  1. 300 000 000 = 3 × 108 m/s. (Move decimal 8 places left.)
  2. 0.000 000 000 053 = 5.3 × 10-11 m. (First non-zero digit is 5; decimal moves 11 places right to give 5.3.)
  3. 1.5 × 1011 = 150 000 000 000 m (1.5 followed by 10 zeros).
  4. Number of cells = 0.014 ÷ (7 × 10-6) = (1.4 × 10-2) ÷ (7 × 10-6) = (1.4/7) × 10(-2-(-6)) = 0.2 × 104 = 2 × 103 (2 000 cells).
  5. 5.97 × 1024 ÷ 7.35 × 1022 = (5.97/7.35) × 102 ≈ 0.8122... × 100 ≈ 81.2. The Earth is 81.2 times heavier (3 s.f.).
  6. Seconds in 1 hour = 3600 = 3.6 × 103. Operations = 2.8 × 109 × 3.6 × 103 = 10.08 × 1012 = 1.008 × 1013 operations.
  7. Total mass = 6.7 × 10-5 × 7.5 × 1018 = (6.7 × 7.5) × 1013 = 50.25 × 1013 = 5.0 × 1014 g (2 s.f.).
  8. Time = distance ÷ speed = (4.01 × 1016) ÷ (3 × 108) = (4.01/3) × 108 = 1.3367 × 108 ≈ 1.34 × 108 seconds (3 s.f.).
  9. 1.5 × 10-7 ÷ 5 × 10-8 = (1.5/5) × 10(-7-(-8)) = 0.3 × 101 = 3 times wider.
  10. Water usage = 8 × 109 × 1250 = 8 × 109 × 1.25 × 103 = 10 × 1012 = 1.0 × 1013 litres (2 s.f.).
Created by Mr Josh