How do you simplify (5.6498times10^10)/(8.2times10^4)?

Jul 6, 2017

$6.9 \times {10}^{5}$

Explanation:

When I simplify fractions, I like to keep track of the terms using parenthesis, so even though it's not the proper way to write scientific notation, I'm going to re-write the problem first.

$\frac{\left(5.6498\right) \left({10}^{10}\right)}{\left(8.2\right) \left({10}^{4}\right)}$

I like to put parenthesis around terms like this because it's easier to see where you can move things around and split the fractions. Not a big deal here, but its a handy trick for larger problems. We can see that if we split the fraction right down the middle;

$\frac{\left(5.6498\right) \textcolor{red}{|} \left({10}^{10}\right)}{\left(8.2\right) \textcolor{red}{|} \left({10}^{4}\right)}$

We can group the $10$s together and the other terms together and solve both parts separately.

$\frac{5.6498}{8.2} \times {10}^{10} / {10}^{4}$

Using long division, the fraction on the left simplifies cleanly to $.689$. To solve the fraction on the right, remember your rules for exponent division. When you divide two numbers with exponents, the exponents subtract.

${10}^{10} / {10}^{4} = {10}^{10 - 4} = {10}^{6}$

Putting both sides back together we have;

$.689 \times {10}^{6}$

To write this properly in scientific notation, we need to move the decimal one place to the right. Remember to account for that in your exponent!

$6.89 \times {10}^{5}$

Lastly, if you are keeping track of significant figures, your smallest term, $8.2 \times {10}^{4}$ has two, so you should round your answer to two significant figures.

$6.9 \times {10}^{5}$