Practice Maths

L31 — Powers of Ten

Key Terms

power (exponent)
The small raised number that tells you how many times to multiply the base by itself. In 10³, the exponent is 3.
base
The number being multiplied. In 10³, the base is 10.
scientific notation
A way of writing very large or small numbers as a value between 1 and 10 multiplied by a power of 10. E.g. 1.5 × 10&sup8;.

Key Rules

Multiplying by a power of 10: move the decimal point right by the number of zeros. E.g. 3.45 × 10² = 345 (2 places right).

Dividing by a power of 10: move the decimal point left by the number of zeros. E.g. 3.45 ÷ 10² = 0.0345 (2 places left).

Worked Example

Calculate 2.56 × 10³ and 4800 ÷ 10²

Step 1 — 2.56 × 10³: move decimal 3 places right → 2560

Step 2 — 4800 ÷ 10²: move decimal 2 places left → 48

Check: 2560 ÷ 10³ = 2.56 ✓   48 × 10² = 4800 ✓

The Place Value System Is Built on Powers of 10

Our number system is called “base 10” — every column in a number is 10 times the column to its right. Thousands → Hundreds → Tens → Ones → Tenths → Hundredths. Each step is a multiplication or division by 10.

When you multiply by 10, all the digits shift one place to the left (the number gets bigger). When you divide by 10, all the digits shift one place to the right (the number gets smaller). The decimal point itself doesn’t move — the digits do!

The Shortcut: Shifting the Decimal Point

In practice, it’s easiest to think of the decimal point shifting:

  • Multiply by 10 → decimal shifts 1 place to the right
  • Multiply by 100 → shifts 2 places right
  • Multiply by 1000 → shifts 3 places right
  • Divide by 10 → decimal shifts 1 place to the left
  • Divide by 100 → shifts 2 places left

Example: 3.45 × 100 = 345 (point shifts 2 right)    3.45 ÷ 100 = 0.0345 (point shifts 2 left)

Count how many zeros are in the power of 10 — that’s how many places the decimal shifts. 1000 has three zeros → shift 3 places.

Index Notation for Powers of Ten

We write powers of 10 using exponents: 10² means 10 × 10 = 100. The exponent tells you the number of zeros after the 1:

  • 10¹ = 10 (one zero)
  • 10² = 100 (two zeros)
  • 10³ = 1,000 (three zeros)
  • 10⁰ = 1 (zero zeros — any number to the power zero equals 1)
  • 10⁶ = 1,000,000 (six zeros — one million)

Real-World Applications

Powers of 10 are everywhere:

  • Computer memory: kilobytes (10³), megabytes (10⁶), gigabytes (10⁹)
  • Distances in astronomy: the Sun is about 1.5 × 10⁸ km away
  • Unit conversions: 1 km = 10³ m, $1 = 10² cents
Common mistake: Shifting the wrong direction. Multiplying makes numbers bigger, so the decimal shifts right. Dividing makes numbers smaller, so it shifts left. Quick check: 5 × 10 should be 50, not 0.5.

  1. Multiply by powers of 10

    1. 5.3 × 10
    2. 5.3 × 100
    3. 5.3 × 1000
    4. 0.72 × 10
    5. 0.72 × 100
    6. 0.72 × 1000

  2. Divide by powers of 10

    1. 620 ÷ 10
    2. 620 ÷ 100
    3. 620 ÷ 1000
    4. 4500 ÷ 10
    5. 4500 ÷ 100
    6. 4500 ÷ 1000

  3. Powers of Ten Values

    What is the value of:

    1. 10³
    2. 10⁴
    3. 10⁰
    4. 10⁵
    5. 10⁶

  4. Multiply using exponent notation

    1. 1.234 × 10²
    2. 0.056 × 10³
    3. 4.5 × 10⁴
    4. 0.007 × 10⁵
    5. 2.8 × 10³

  5. Divide using exponent notation

    1. 3560 ÷ 10³
    2. 90 ÷ 10²
    3. 48000 ÷ 10⁴
    4. 7200 ÷ 10³
    5. 500 ÷ 10²

  6. Express in standard decimal form

    1. 3.2 × 10⁵
    2. 7.89 × 10²
    3. 6.02 × 10⁵
    4. 3.75 × 10³
    5. 1.5 × 10⁴

  7. Factory Production

    A factory produces 1.5 × 10³ items per day. How many items are produced in 10 days?

  8. Stadium Capacity

    A stadium seats 4.5 × 10⁴ people. If 10 stadiums have the same capacity, what is the total? Write your answer using scientific notation.

  9. Which Power?

    Fill in the missing power of 10:

    1. 3.4 × ___ = 3400
    2. 560 ÷ ___ = 0.56
    3. 0.09 × ___ = 90
    4. 12000 ÷ ___ = 12

  10. Distance to the Sun

    The distance from Earth to the Sun is approximately 1.5 × 10⁸ km. Write this in full as a decimal number. If a spaceship travels at 1.5 × 10⁴ km/h, how many hours would the trip take?

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