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Z.2 Use exponents to multiply decimals by powers of ten

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What does it mean to use exponents with powers of ten?

Exponents are a way to show how many times a number is multiplied by itself. When we use powers of ten, the exponent tells us how many zeros the number ten has or how many times we move the decimal point.

Examples:
  • 101 = 10
  • 102 = 100
  • 103 = 1,000
Note

The exponent shows how many places the decimal point moves. A positive exponent moves the decimal to the right, and a negative exponent moves it to the left.

How to multiply a decimal by powers of ten using exponents

When you multiply a decimal by a power of ten written with an exponent, move the decimal point to the right the same number of places as the exponent shows.

Example:
  • 3.45 × 102 = move the decimal 2 places to the right → 345
  • 0.76 × 101 = move the decimal 1 place to the right → 7.6
Note

Each increase in the exponent makes the number ten times larger. Be sure to count the decimal places carefully when moving the decimal point.

Multiplying decimals by negative exponents

A negative exponent means dividing by a power of ten. When you multiply a decimal by ten raised to a negative exponent, move the decimal point to the left.

Example:
  • 4.8 × 10−1 = move the decimal 1 place to the left → 0.48
  • 6.25 × 10−2 = move the decimal 2 places to the left → 0.0625
Note

Negative exponents make numbers smaller because you are dividing by ten instead of multiplying. Remember: the negative sign in the exponent means “move the decimal left.”

Connecting exponents and place value

Exponents and place value are closely related. Each time you multiply or divide by ten, you move to the next place value. Exponents simply show how many times this happens.

Example:
  • 2.5 × 103 → move 3 places right → 2,500
  • 2.5 × 10−3 → move 3 places left → 0.0025
Note

Understanding exponents helps you see patterns in place value. Each jump to the right or left changes the number’s size by a factor of ten.

Check your understanding

Before moving on, make sure you can explain what an exponent means and how to multiply a decimal by a power of ten using exponents.

Quick Practice:
  • 5.2 × 101 = ?
  • 7.09 × 102 = ?
  • 3.6 × 10−1 = ?
Note

If you can move the decimal point correctly and understand what each exponent means, you are ready to apply this skill to more advanced problems.