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C.1 Understand powers of ten

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What are powers of ten?

Powers of ten show how many times the number ten is used as a factor. Each power of ten represents multiplying ten by itself a certain number of times.

Examples:
  • 10¹ = 10 (ten multiplied once)
  • 10² = 10 × 10 = 100
  • 10³ = 10 × 10 × 10 = 1,000
Note

The exponent tells how many times you multiply ten by itself. A higher exponent makes the number grow quickly.

Understanding exponents

An exponent shows how many times the base number is multiplied by itself. In powers of ten, the base is always 10, and the exponent tells you how many zeros the number has.

Example:
  • 10⁴ = 10 × 10 × 10 × 10 = 10,000
  • 10⁵ = 10 × 10 × 10 × 10 × 10 = 100,000
Note

The number of zeros in the product is the same as the exponent. For example, 10³ has three zeros: 1,000.

Multiplying and dividing by powers of ten

When you multiply a number by a power of ten, move the digits to the left. When you divide by a power of ten, move the digits to the right.

Examples:
  • 24 × 10 = 240 (move digits 1 place left)
  • 24 × 100 = 2,400 (move digits 2 places left)
  • 24 ÷ 10 = 2.4 (move digits 1 place right)
Note

Each time you multiply or divide by 10, the value of each digit changes its place value. Multiplying increases the size; dividing decreases it.

Writing powers of ten in standard and exponential form

Numbers that are multiples of ten can be written in two ways: standard form (with zeros) and exponential form (with exponents).

Examples:
  • 10 × 10 × 10 = 1,000 → standard form: 1,000 → exponential form: 10³
  • 10 × 10 × 10 × 10 × 10 = 100,000 → exponential form: 10⁵
Note

Exponential form is a shorter way to show repeated multiplication by 10. It is often used in science and large number calculations.

Understanding powers of ten with decimals

Negative exponents show powers of ten that are smaller than one. Each negative power moves the digits to the right, dividing by ten each time.

Examples:
  • 10⁻¹ = 1 ÷ 10 = 0.1
  • 10⁻² = 1 ÷ 100 = 0.01
  • 10⁻³ = 1 ÷ 1,000 = 0.001
Note

Each negative exponent means the decimal point moves one place to the left for each step down in power.