As researchers work with scientific notation, they may need to multiply numbers or divide them. This section will show you how to do that.
Example 1: Multiplying Numbers in Scientific Notation
Simplify and express the answer in scientific and standard notation.
(3.2 x 10-2) (5.6 x 1015)
All terms in this expression, (3.2 x 10-2) (5.6 x 1015), are being multiplied. That means that you can use the commutative and associative properties to regroup the numbers.
(3.2 x 10-2) (5.6 x 1015)
= (3.2 x 5.6) (10-2 x 1015)
Regroup the numbers so that the powers of 10 are together.
= (17.92) (10-2 x 1015)
Multiply the numbers in the first parenthesis.
= 17.92 x 1013
Add the exponents of base 10.
= (1.792 x 101) x 1013
= 1.792 x 1014
Scientific Notation
Move the decimal just after the 1 in 17.92. Since 17.92 > 0, add a positive one to the exponent for the one place moved.
=179,200,000,000,000
Standard Notation
(5 x 105) (3.1 x 10-10)
All terms in this expression are being multiplied. That means that you can use the commutative and associative properties to regroup the numbers.
(5 x 105) (3.1 x 10-10)
= (5 x 3.1) (105 x 10-10)
Regroup the numbers so that the powers of 10 are together.
= (15.5) (105 x 10-10)
Multiply the numbers in the first parenthesis.
= 15.5 x 10-5
Add the exponents of base 10.
= (1.55 x 101) x 10-5
= 1.55 x 10-4
Scientific Notation
Move the decimal just after the 1 in 15.5. Since 15.5 > 0, add a positive one to the exponent for the one place moved.
= 0.000155
Standard Notation
Example 2: Dividing Numbers in Scientific Notation
Evaluate the following and express the solution in scientific and standard notation.
All terms in this expression are being multiplied. That means that you can use the commutative and associative properties again to regroup the numbers.
Regroup the numbers and bases.
Divide in the first fraction.
Subtract the exponents of base 10. Remember that you subtract exponents when you divide common bases.
Scientific Notation
Standard Notation
Regroup the numbers and bases.
Divide in the first fraction.
Subtract the exponents of base 10. Remember that you subtract exponents when you divide common bases.
Scientific Notation
Standard Notation
Practice
Evaluate each of the following. Leave your answer in scientific and standard notation.
Problem
Answer
Scientific Notation
Standard Notation
(4.9 x 104) (4 x 103)
19.6 x 107
1.96 x 108
196,000,000
(5 x 106)(3.2 x 102)
16 x 108
1.6 x 109
1,600,000,000
(8.6 x 10-2)(4 x 10-3)
34.4 x 10-5
3.44 x 10-4
0.000344
(4.8 x 10-2)(1.8 x 10-2)
8.64 x 10-4
8.64 x 10-4
0.000864
(2.5 x 107)(6.3 x 10-9)
15.75 x 10-2
1.575 x 10-1
0.1575
9 x 105
9 x 105
900,000
2.2 x 10-4
2.2 x 10-4
0.00022
0.89 x 10-6
8.9 x 10-5
0.000089
1.19 x 10-4
1.19 x 10-4
0.000119
0.23 x 10-5
2.3 x 10-6
0.0000023
Section 1 Homework (10 points)
It’s time to complete your homework. Find the section homework link to submit for a grade. You may take this only one time, so check your understanding of the material before completing your homework, and do your best.
Practice
Section 1 Homework (10 points)
