Basic Mathematics for College Students

4.3 Multiplying Decimals 347

Multiplying a Decimal by 10, 100, 1,000, and So On

To find the product of a decimal and 10, 100, 1,000, and so on, move the decimal

point to the right the same number of places as there are zeros in the power of 10.

2 Multiply decimals by powers of 10.

The numbers 10, 100, and 1,000 are called powers of 10,because they are the results when
we evaluate 10^1 ,10^2 , and 10^3. To develop a rule to find the product when multiplying a
decimal by a power of 10, we multiply 8.675 by three different powers of 10.

Multiply: 8.675  10 Multiply: 8.675  100 Multiply: 8.675 1,000

When we inspect the answers, the decimal point in the first factor 8.675 appears
to be moved to the right by the multiplication process. The number of decimal places
it moves depends on the power of 10 by which 8.675 is multiplied.

One zero in 10 Two zeros in 100 Three zeros in 1,000

8.675 10 86.75 8.675 100 867.5 8.6751,000 8675

8.675

 1000

0000

00000

000000

8675000

8675.000

8.675

 100

0000

00000

867500

867.500

8.675

 10

0000

86750

86.750

EXAMPLE (^4) Multiply: a.2.81  10 b.0.076(10,000)
StrategyFor each multiplication, we will identify the factor that is a power of 10,
and count the number of zeros that it has.
WHYTo find the product of a decimal and a power of 10 that is greater than 1, we
move the decimal point to the right the same number of places as there are zeros
in the power of 10.
Solution
a.2.81  10 28.1 Since 10 has 1 zero, move the decimal point 1 place to the right.
b. 0.076(10,000) 0760. Since 10,000 has 4 zeros, move the decimal point 4 places

Self Check 4
Multiply:
a.0.721  100
b.6.08(1,000)
Now TryProblems 21 and 23
Numbers such as 10, 100, and 1,000 are powers of 10 that are greater than 1.There
are also powers of 10 that are less than 1,such as 0.1, 0.01, and 0.001. To develop a rule
to find the product when multiplying a decimal by one tenth, one hundredth, one
thousandth, and so on, we will consider three examples:
Multiply: 5.19 0.1 Multiply: 5.19 0.01 Multiply: 5.19 0.001
  

5.19

 0.001

0.00519

5.19

 0.01

0.0519

5.19

 0.1

0.519

It moves 1 place It moves 2 places It moves 3 places

to the right. to the right. to the right.

These observations illustrate the following rule.

  

to the right. Write a placeholder zero (shown in blue).

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