Multiply Decimals by Powers of Ten | 5th Grade Math

✖️ Multiply Decimals by Powers of Ten

Grade 5 Math - Complete Notes & Formulae

Key Concepts

  • Multiplying a decimal by a power of ten moves the decimal point to the right.
  • Powers of ten: \(10, 100, 1000, 10^n\), etc.
  • Multiplying by 0.1, 0.01: Moves the decimal point to the left.
  • Exponents: \(10^n\) means multiplying by 10, \(n\) times.

1️⃣ Multiplying by Powers of Ten (\(10, 100, 1000\), etc.)

Rule: For each zero in the power of ten, move the decimal point one place to the right.
  • Number of zeros = number of places to move.
  • \(a.bc \times 10 = ab.c\)
  • \(a.bc \times 100 = abc.\)
  • \(a.bc \times 1000 = abcd.\)
With exponents:
  • \(a.bc \times 10^n\) → move the decimal \(n\) places to the right.
  • If there are not enough digits, add zeros.

Examples:

  1. \(2.3 \times 10 = 23\)
  2. \(0.67 \times 100 = 67\)
  3. \(5.429 \times 1000 = 5429\)
  4. \(0.51 \times 10^2 = 0.51 \times 100 = 51\)
Shortcut: Just move the decimal point!

Main Formula

\(x.yz \times 10^n = xy.z\) (move decimal point \(n\) places right)

2️⃣ Multiplying by 0.1, 0.01, or Other Fractional Powers

Rule: For each zero after the decimal point, move the decimal point one place to the left.
  • \(a.bc \times 0.1 = 0.abc\)
  • \(a.bc \times 0.01 = 0.0abc\)
  • \(a.bc \times 0.001 = 0.00abc\)

Examples:

  1. \(9.3 \times 0.1 = 0.93\)
  2. \(0.67 \times 0.1 = 0.067\)
  3. \(5.429 \times 0.01 = 0.05429\)
  4. \(0.51 \times 0.01 = 0.0051\)
Shortcut: Just move the decimal point left for each decimal place.

Main Formula

\(x.yz \times 0.1^n = 0.xy...z\) (move decimal \(n\) places left)

3️⃣ Using Exponents

  • Exponential form: \(10^n\) is a shortcut for multiplying by 10, several times.
  • Examples: \(10^2=100\), \(10^3=1000\)
  • Multiply as before: move decimal \(n\) places to the right for \(10^n\).

Example (with exponents):

  1. \(0.437 \times 10^3 = 437\)
  2. \(0.59 \times 10^2 = 59\)
  3. \(1.93 \times 10^1 = 19.3\)

Exponent Rule

\(x.yz \times 10^n = xy.z\) (move decimal \(n\) places right)

4️⃣ Find the Missing Number

  • If \(x \times 10 = 123.4\), what is \(x\)?
    Move decimal one place left: \(x = 12.34\)
  • If \(x \times 1000 = 7.14\), what is \(x\)?
    Move decimal three places left: \(x = 0.00714\)
  • If \(54.9 = x \times 0.1\), what is \(x\)?
    Move decimal one place right: \(x = 549\)

5️⃣ Visual Explanation

Decimal Place Value Table

ThousandsHundredsTensOnesDecimalTenthsHundredthsThousandths
1,000100101.0.10.010.001
Multiplying by ten shifts all digits one place LEFT; by 0.1 shifts one place RIGHT!

6️⃣ Practice Problems

  1. \(4.73 \times 100\) = ______
  2. \(0.56 \times 10^2\) = ______
  3. \(8.12 \times 0.1\) = ______
  4. \(0.763 \times 10^3\) = ______
  5. \(5.09 \times 0.01\) = ______
  6. \(x \times 10 = 52.4\). What is \(x\)?
Answers:
1) 473
2) 56
3) 0.812
4) 763
5) 0.0509
6) 5.24

💡 Tip: The number of zeros tells you how many places to move the decimal! Practice to master these skills!