# How do you simplify (8.39 xx 10^-7)(4.53 xx 10^9)?

Mar 19, 2018

$\left(8.39 \times {10}^{-} 7\right) \times \left(4.53 \times {10}^{9}\right)$

$= 3.80067 \times {10}^{3}$

#### Explanation:

Consider the product of $\text{ } 3 {x}^{-} 7 \times 7 {x}^{9}$

$\rightarrow$ multiply the numbers and add the indices of like bases.

$= 21 {x}^{- 7 + 9}$

$= 21 {x}^{2}$

Exactly the same happens when you are multiplying numbers in scientific notation.

$\rightarrow$ multiply the numbers and add the indices of like bases.

$\left(8.39 \times {10}^{-} 7\right) \times \left(4.53 \times {10}^{9}\right)$

$38.0067 \times {10}^{- 7 + 9}$

$\textcolor{b l u e}{38} .0067 \times {10}^{2} \text{ } \leftarrow$ not in scientific notation.

$= 3.80067 \times {10}^{3}$

Mar 19, 2018

See a solution process below:

#### Explanation:

First, rewrite the expression as:

$\left(8.39 \times 4.53\right) \times \left({10}^{-} 7 \times {10}^{9}\right) \implies$

$38.0067 \times \left({10}^{-} 7 \times {10}^{9}\right)$

Now, we can use this rule for exponents to simplify the 10s terms:

${x}^{\textcolor{red}{a}} \times {x}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} + \textcolor{b l u e}{b}}$

$38.0067 \times \left({10}^{\textcolor{red}{- 7}} \times {10}^{\textcolor{b l u e}{9}}\right) \implies$

$38.0067 \times {10}^{\textcolor{red}{- 7} + \textcolor{b l u e}{9}} \implies$

$38.0067 \times {10}^{2}$

To write this in scientific notation we need to move the decimal point one place to the left so we need to add $1$ to the 10s exponent:

$3.80067 \times {10}^{2 + 1} \implies$

$3.80067 \times {10}^{3}$

Or, in standard terms:

$3800.67$

Mar 19, 2018

By first combining the exponential terms, you can simplify the problem significantly, and get $3.80067 \times {10}^{3}$ as the result.

#### Explanation:

The first thing to do is rewrite the function so all of the multiplication is outside of parentheses, and the exponents are grouped together:

$\left(8.39 \times {10}^{- 7}\right) \left(4.53 \times {10}^{9}\right) = 8.39 \times 4.53 \times {10}^{9} \times {10}^{- 7}$

Next, lets multiply the constants together, and use scientific notation to describe the result:

$8.39 \cdot 4.53 = 38.0067 = 3.80067 \times {10}^{1}$

Re-writing the equation:

$3.80067 \times {10}^{1} \times {10}^{9} \times {1010}^{- 7}$

Finally, we combine the notation terms. When you multiply two exponential numbers with the same base, it is the same as adding the exponents together, like so:

${x}^{n} \times {x}^{z} = {x}^{n + z}$

using that principle:

${10}^{1} \times {10}^{9} \times {10}^{- 7} = {10}^{1 + 9 - 7} = {10}^{3}$

Now, lets re-write the equation one last time with all of our simplification:

$\textcolor{red}{3.80067 \times {10}^{3}}$

Mar 19, 2018

Suppose the idea of decimals is giving you a problem.

$3.80067 \times {10}^{3}$

#### Explanation:

$\textcolor{b l u e}{\text{Consider: } 8.39}$
this is the same as $839 \times \frac{1}{100} = 839 \times {10}^{- 2}$

So we have:

$\left[8.39\right] \times {10}^{- 7} \to \left[839 \times {10}^{- 2}\right] \times {10}^{- 7} = 839 \times {10}^{- 9}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\textcolor{b l u e}{\text{Consider: } 4.53}$

Using the same argument as above we have:

$\left[453 \times {10}^{- 2}\right] \times {10}^{9} = 453 \times {10}^{7}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Putting it all together}}$

$\left(839 \times {10}^{- 9}\right) \left(453 \times {10}^{7}\right) = 839 \times 453 \times {10}^{7} / {10}^{9}$

$380067 \times \frac{1}{10} ^ 2$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$380067 \times \frac{1}{10} \times \frac{1}{10} = 3800.67$

$3800.67 \text{ is the same as } 380.067 \times 10$
$3800.67 \text{ is the same as } 38.0067 \times 10 \times 10$
$3800.67 \text{ is the same as } 3.80067 \times 10 \times 10 \times 10$

So in Scientific notation we have: $3.80067 \times {10}^{3}$