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

Jun 28, 2016

$2.7 \cdot {10}^{-} 11$

#### Explanation:

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

$\left(3.26 \cdot {10}^{-} 6\right) \left(8.2 \cdot {10}^{-} 6\right) = 3.26 \cdot {10}^{-} 6 \cdot 8.2 \cdot {10}^{-} 6$

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

$\left(3.26 \cdot 8.2\right) \cdot \left({10}^{-} 6 \cdot {10}^{-} 6\right)$

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

${a}^{b} \cdot {a}^{c} = {a}^{b + c}$

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

${10}^{-} 6 \cdot {10}^{-} 6 = {10}^{- 6 + \left(- 6\right)} = {10}^{-} 12$

Simplifying, we obtain:

$26.732 \cdot {10}^{-} 12$

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.

$26.732 \cdot {10}^{-} 12 = 2.6732 \cdot {10}^{-} 11$

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,

$2.6732 \cdot {10}^{-} 11 \to 2.7 \cdot {10}^{-} 11$