Lesson Objectives
• Demonstrate an understanding of decimals
• Demonstrate an understanding of multiplication with whole numbers
• Learn how to quickly multiply a decimal by 10 or a power of 10
• Learn how to multiply two or more decimals

## How to Multiply Decimals

We previously learned how to multiply multi-digit whole numbers. We will build on this process and show how to find the product of two or more decimals. When we multiply two or more decimals together:
• Ignore all decimal points in each factor (numbers being multiplied) » pretend the numbers are whole numbers
• Find the product of the whole numbers and place a visible decimal point after the final digit (rightmost) of the answer
• Count the total number of decimal places (places to the right of the decimal point) between all factors
• Move the decimal point in the answer to the left by the number of decimal places counted in the last step
Let's take a look at a few examples.
Example 1: Find each product.
3.25 x 7.17
Ignore the decimal points in each factor and find the product:
325 x 717 = 233,025
Place a visible decimal point after the final digit of the number:
233,025.
Count the total number of decimal places between all factors:
3.25 » (2 decimal places)
7.17 » (2 decimal places)
Between the two factors, we have a total of four decimal places.
We will move our decimal point in the number (233,025.) four places to the left:
23.3025
3.25 x 7.17 = 23.3025
Example 2: Find each product.
4.5 x 1.02 x 0.009
Ignore the decimal points in each factor and find the product:
45 x 102 x 9 = 41,310
Place a visible decimal point after the final digit of the number:
41,310.
Count the total number of decimal places between all factors:
4.5 » (1 decimal place)
1.02 » (2 decimal places)
0.009 » (3 decimal places)
Between the three factors, we have a total of six decimal places.
We will move our decimal point in the number (41,310.) six places to the left:
0.041310
4.5 x 1.02 x 0.009 = 0.04131

### Multiplying by 10 or a Power of 10

When we multiply by 10 or a positive multiple of 10 (10, 100, 1000,...) we can use a shortcut. Observe the following pattern:
5 x 1 = 5
5 x 10 = 50
5 x 100 = 500
5 x 1000 = 5000
At this point, we know the trick for multiplying when we have powers of ten or trailing zeros. We learned that we could multiply the non-zero numbers and attach the total number of trailing zeros between all factors to the end of the number. When we multiply by 10 or a power of 10, we can just move the decimal point one place to the right for each 0 in the power of 10. We previously learned that to the right of a decimal point, we can add zeros after the final non-zero digit and not change the value of a number. Let's look at an example. Suppose we start with the number 3:
We can write 3 as 3.0:
3.0
If we multiply by 10, we have one zero in the power of 10, we can just move the decimal point one place to the right to obtain our answer.
3.0 x 10 = 30.0
Suppose we tried the same technique with 10,000. This power of 10 has four zeros:
3 x 10,000
3.0 x 10,000
We move our decimal point four places to the right:
3.0 x 10,000 = 30,000.0
Example 3: Find each product.
17.215 x 1000
We are multiplying by 1000, this power of 10 has three zeros:
17.215 x 1000
We move our decimal point three places to the right:
17,215.0
Example 4: Find each product.
1.907516 x 10,000
We are multiplying by 10,000, this power of 10 has four zeros:
1.907516 x 10,000
We move our decimal point four places to the right:
19,075.16
When we see a power of 10 in exponent form, we can use the same process. Recall that a power of 10 such as 103 can be evaluated quickly by writing a 1 followed by the exponent number of zeros:
103 = 1000
107 = 10,000,000
When we see a decimal multiplied by a power of 10 in exponent form, we move our decimal point to the right by the same number of digits as the exponent on 10. Let's look at a few examples.
Example 5: Find each product.
3.99156 x 104
Our exponent on 10 is a 4.
We move our decimal point in the number 3.99156 four places to the right:
3.99156 » 39,915.6
3.99156 x 104 = 39,915.6
Example 6: Find each product.
0.228195001 x 108
Our exponent on 10 is an 8.
We move our decimal point in the number 0.228195001 eight places to the right:
0.228195001 » 22,819,500.1
0.228195001 x 108 = 22,819,500.1

#### Skills Check:

Example #1

Find each product.

-1.61 x 9.43 x (-4.9)

A
-74.39327
B
74.39327
C
69.8513
D
-59.27157
E
70.68751

Example #2

Find each product.

3.01 x (-7.4) x 9.3

A
-199.2469
B
-64.3814
C
-101.5981
D
-301.5687
E
-207.1482

Example #3

Find each product.

-6.8 x (-3.664)

A
24.9152
B
22.2132
C
18.9164
D
3.5514
E
37.8912