Question

How do you simplify and write `(3.26times10^-6)(8.2times10^-6)` in standard form?

Answer 1

`2.7 * 10^-11`

Explanation:

First, recognize that the expression involves the multiplication of four terms. Because of this, the parentheses can be removed.

`(3.26 *10^-6)(8.2*10^-6)=3.26*10^-6*8.2*10^-6`

Next, use the commutative property of multiplication to group the terms that are base 10 and the terms that are not base 10.

`(3.26*8.2)*(10^-6*10^-6)`

When terms containing like bases are multiplied together, their exponents will be added together. Mathematically, this is shown as:

`a^b*a^c=a^(b+c)`

Using this, the base-10 terms can be simplified:

`10^-6*10^-6=10^(-6+(-6))=10^-12`

Simplifying, we obtain:

`26.732*10^-12`

However, in the standard form of scientific notation, the number preceding the base-10 term should always be between 1 and 10. To do this, move the decimal of the 26.732 term to the left by one place, and increase the exponent of the `10^-12` term by one.

`26.732*10^-12=2.6732*10^-11`

Finally, since the number with the smallest number of significant figures was 8.2 in the original multiplication, the final answer should have 2 significant figures as well. Hence,

`2.6732*10^-11->2.7*10^-11`