# Question #faff2

Oct 7, 2016

$\frac{a \times {10}^{n}}{b \times {10}^{m}} = \frac{a}{b} \times {10}^{n - m}$

#### Explanation:

Using the property of exponents that ${x}^{a} / {x}^{b} = {x}^{a - b}$, suppose $a \times {10}^{n}$ and $b \times {10}^{m}$ are two numbers in scientific notation (i.e. $a , b \in \left[1 , 10\right)$) Then

$\frac{a \times {10}^{n}}{b \times {10}^{m}} = \frac{a}{b} \times {10}^{n} / {10}^{m} = \frac{a}{b} \times {10}^{n - m}$

Note that $\frac{1}{10} < \frac{a}{b} < 10$, meaning an additional ${10}^{- 1}$ may be multiplied ${10}^{n - m}$ if we are writing the answer in scientific notation.

Some examples:

$\frac{6 \times {10}^{5}}{2 \times {10}^{9}} = \frac{6}{2} \times {10}^{5 - 9} = 3 \times {10}^{-} 4$

$\frac{2 \times {10}^{7}}{5 \times {10}^{3}} = \frac{2}{5} \times {10}^{7 - 3} = 0.4 \times {10}^{4} = 4 \times {10}^{3}$