Multiplying Decimals by Powers of 10: Fifth Grade Mathematics

Multiplying decimals by powers of 10 shifts the decimal point to the right, reflecting place value growth by factors of ten.

The Direction and Mechanics of Decimal Movement

When a number is multiplied by 10, each digit increases in value by a factor of 10, shifting one place value to the left. Visually, this is represented by moving the decimal point one position to the right.

+--------------------------------------------------------------------------+
|                  DECIMAL SHIFTS WITH POWERS OF TEN                       |
|                                                                          |
|   Multiplying by 10^1 (10):      Shift decimal point 1 place RIGHT       |
|   Multiplying by 10^2 (100):     Shift decimal point 2 places RIGHT      |
|   Multiplying by 10^3 (1,000):   Shift decimal point 3 places RIGHT      |
|   Multiplying by 10^4 (10,000):  Shift decimal point 4 places RIGHT      |
|                                                                          |
|   Rule: Number of zeros in the power of 10 = Number of places shifted.   |
+--------------------------------------------------------------------------+

Worked Progression

Observe how 3.425 transforms as it is multiplied by increasing powers of ten:

  3.425 * 10^1  =  3.425 * 10      =   34.25       (1 place right)
  3.425 * 10^2  =  3.425 * 100     =  342.5        (2 places right)
  3.425 * 10^3  =  3.425 * 1,000   = 3425.         (3 places right = 3,425)
  3.425 * 10^4  =  3.425 * 10,000  = 34250.        (4 places right = 34,250)

Annexing Zeros When Decimal Places Run Out

When the power of ten requires shifting more places than there are decimal digits available, annex zeros as placeholders. In 0.6 * 10^3: shifting 1 place right gives 6; shifting 2 more places right requires annexing two zeros, giving 600.

Multiplying 0.48 * 10^4:
  Original:      0 . 4 8
  Shift 1:       4 . 8
  Shift 2:      4 8 .
  Shift 3:     4 8 0 .
  Shift 4:   4 8 0 0 .
  Result: 4,800

Connecting Exponents to Place Value Growth

The exponent on base 10 directly tells you the number of shifts. In 7.82 * 10^3, the exponent 3 signals an increase of three place values (10 * 10 * 10 = 1,000), shifting the decimal point 3 places to the right to produce 7,820.

Exercises and Solutions

Exercise 1: Mental Shifts with Exponents. Solve the following expressions mentally by applying decimal point shift rules: Part A: 0.084 * 10^2. Part B: 15.6 * 10^3.

Solution 1: For Part A, multiplying by 10^2 means multiplying by 100, which shifts the decimal point 2 places to the right. Moving 0.084 two places right gives 8.4. For Part B, multiplying by 10^3 means multiplying by 1,000, which shifts the decimal point 3 places to the right. Moving 15.6 one place right gives 156, and moving two additional places right requires annexing two zeros, producing 15,600.

Exercise 2: Metric Measurement Conversion. A laboratory technician has a sample of liquid measuring 0.375 liters. To convert liters to milliliters, the quantity must be multiplied by 1,000 (10^3). Calculate the volume in milliliters and explain the shift.

Solution 2: Multiply 0.375 by 10^3: 0.375 * 1,000. Since 1,000 has 3 zeros (exponent 3), shift the decimal point 3 places to the right: 0.375 becomes 3.75 after 1 shift, 37.5 after 2 shifts, and 375 after 3 shifts. The volume of the sample is 375 milliliters.