# Multiplying Numbers in Scientific Notation

Numbers expressed in scientific notation are multiplied with the help of the commutative and associative properties of exponents.

• According to the commutative law, a + b = b + a
• According to the associative law, a + (b + c) = (a + b) + c

Here are the steps to follow while multiplying numbers in scientific notation.

## Steps with Examples

1. Separate and divide the coefficient and exponent parts separately. Rearrange the numbers using the commutative and associative properties of exponents.
2. Multiply the bases using the multiplication rule of exponents (am Ã— an = am + n), and thus the exponents of the denominator are added from the numerator.
3. Combine the new coefficient with the new power of 10 to get the answer. If the product of the coefficients is greater than 9, convert it to scientific notation and multiply by 101.

With Positive Exponents

Example – 1: Multiply and express in scientific notation: (9 x 106) Ã— (3 x 103)

1. Rearranging the numbers using the commutative and associative properties of exponents

= (9 Ã— 3) Ã— (106 Ã— 103)

• Multiplying the coefficients and using the product rule of exponents

= 9 Ã— 3 = 27, 106 Ã— 103 = 106 + 3 = 109

• The product of the coefficients is 27 and is greater than 9, therefore converting it again to scientific notation and multiplying by 101

= (27 Ã— 101) x 109

= 27 Ã— 1010

Thus, the answer is 27 Ã— 1010

With Negative Exponent

Example – 2: Multiply and express in scientific notation: (9.2 x 106) Ã— (1.5 x 10-2)

1. Rearranging the numbers using the commutative and associative properties of exponents

= (9.2 Ã— 1.5) Ã— (106 Ã— 10-2)

• Multiplying the coefficients and using the product rule of exponents

= 9.2 Ã— 1.5 = 13.8, 106 Ã— 10-2 = 106 + (-2) = 104

• The product of the coefficients is 13.8 and is greater than 9, therefore converting it again to scientific notation by moving the point one place to the right and multiplying by 101

= (13.8 Ã— 101) x 104

= 1.38 Ã— 105

Thus, the answer is 1.38 Ã— 105

Let us solve some problems.

Multiply: (3.3 x 105) x (2.65 x 103)

Solution:

Rearranging the numbers using the commutative and associative properties of exponents
= (3.3 Ã— 2.65) Ã— (105 Ã— 103)
Multiplying the coefficients and using the product rule of exponents
= 9.2 Ã— 3.5 Ã— 1.5 = 8.745, 105 Ã— 103 = 105 + 3 = 108
The product of the coefficients is 8.745 and is lesser than 9, thus multiplying the coefficient and the new power of 10, we get
= 8.745 Ã— 108
Thus, the answer is 8.745 Ã— 108

Evaluate: (6.2 Ã— 106) (3.5 Ã— 10-3) (1.5 Ã— 10-7)

Solution:

Rearranging the numbers using the commutative and associative properties of exponents
= (6.2 Ã— 3.5 Ã— 1.5) Ã— (106 Ã— 10-3 Ã— 10-7)
Multiplying the coefficients and using the product rule of exponents
= 9.2 Ã— 3.5 Ã— 1.5 = 48.3, 106 Ã— 10-3 Ã— 10-7 = 106 + (-3) + (-7) = 104
The product of the coefficients is 48.3 and is greater than 9, therefore converting it again to scientific notation by moving the point one place to the right and multiplying by 101
= (48.3 Ã— 101) x 104
= 4.38 Ã— 105
Thus, the answer is 4.38 Ã— 105

Multiplying a WHOLE NUMBER by a Scientific Notation

Multiply: 4 Ã— (3 Ã— 104)

Solution:

Rearranging the numbers using the commutative and associative properties of exponents, keeping the exponential part as it is
= (4 Ã— 3) Ã— (104)
Multiplying the coefficients
= (4 Ã— 3)
= 12
Multiplying the coefficient and the exponential part, we get
= 12 Ã— 104
Thus, the answer is 12 Ã— 104