Method

Multiply Numbers in Scientific Notation

Numbers written in scientific notation can be multiplied by using the properties of exponents. Two numbers written in scientific notation a* 10^b and c * 10^d can be multiplied by using the Commutative Property of Multiplication and the Product of Powers Property.

( a* 10^b) * ( c* 10^d)= a c * 10^(b+d)

The first factors of the numbers are multiplied like integers or decimal numbers. Then, the exponents of the second factors are added. Since they have the same base 10, the Product of Powers Property can be used. As an example, consider the following product. (1.5 * 10^2)*(12 * 10^5) Three steps can be followed to multiply these numbers.

Write Each Number in Scientific Notation

It is helpful to check whether both of the numbers are written in scientific notation. Recall that a number is in scientific notation if it is written as a product of two numbers. One of these numbers should be a power of 10. The other number should be greater than or equal to 1 and less than 10. c First Factor & * & Second Factor 1.5 & * & 10^2 & ✓ 12 & * & 10^5 & * The first number is in scientific notation. The second number is not. Its first factor 12 is greater than 10. Note that 12 can also be expressed as 12.0. The decimal point needs to move one unit to the left for the first factor to become less than 10. The second factor's power of 10 is then increased by that number of moves, 1. 12 * 10^5 ⇔ 1.2 * 10^(5+1) ⇔ 1.2*10^6 ✓

Multiply the First Factors

Both numbers are now written in scientific notation. The first factors of both numbers can be multiplied by using the Commutative Property of Multiplication.

(1.5 * 10^2)*(1.2* 10^6)

RemovePar

Remove parentheses

1.5 * 10^2 * 1.2* 10^6

CommutativePropMult

Commutative Property of Multiplication

1.5 * 1.2 * 10^2 * 10^6

Multiply

1.8 * 10^2 * 10^6

Multiply the Second Factors

Next, the powers with base 10 are multiplied. Remember to use the Product of Powers Property — exponents are added when multiplying powers with the same base.

1.8 * 10^2 * 10^6

MultPow

a^m*a^n=a^(m+n)

1.8 * 10^8

Write the Result in Scientific Notation

The result is 1.8 * 10^8. The first factor is greater than 1 and less than 10. The second factor is written as a power of 10. Therefore, the result is already in scientific notation.

Recommended exercises