# How do you express 90000/0.03 in scientific notation and then divide using the rules of exponents?

Jun 18, 2018

$3 \cdot {10}^{6}$

#### Explanation:

In scientific notation, you have $90000 = 9 \cdot {10}^{4}$ and $0.03 = 3 \cdot {10}^{- 2}$. The fraction becomes

$\setminus \frac{9 \cdot {10}^{4}}{3 \cdot {10}^{- 2}}$

We can simplify the numeric and exponential parts separately, so we can begin with

$\setminus \frac{{\cancel{9}}^{3} \cdot {10}^{4}}{\cancel{3} \cdot {10}^{- 2}} = \setminus \frac{3 \cdot {10}^{4}}{{10}^{- 2}}$

As for the exponential part, we use the definitions

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

${a}^{b} / {a}^{c} = {a}^{b - c}$

to conclude that

$\setminus \frac{3 \cdot {10}^{4}}{{10}^{- 2}} = 3 \cdot {10}^{4 - \left(- 2\right)} = 3 \cdot {10}^{6}$

Jun 18, 2018

$3.0 \times {10}^{6}$

#### Explanation:

Given: $\frac{90000}{0.03}$

$\textcolor{b r o w n}{\text{ Both methods}}$

$\textcolor{b l u e}{\text{Without the rules of exponents for comparison}}$

Lets get rid of the decimal.

$\frac{90000}{0.03} \times 1 \textcolor{w h i t e}{\text{d")->color(white)("d}} \frac{90000}{0.03} \times \frac{100}{100} = \frac{9000000}{3}$

Giving: $3000000 = 3.0 \times {10}^{6}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Using rules of exponents}}$

$\textcolor{p u r p \le}{\frac{90000}{0.03} \textcolor{w h i t e}{\text{d") = color(white)("d")(9xx10^4)/(3xx10^(-2))color(white)("d") =color(white)("d}} \frac{9}{3} \times \frac{{10}^{4}}{{10}^{- 2}}}$

but 1/(10^("negative 2")) = 10^("positive 2") giving:

$\textcolor{p u r p \le}{\frac{9}{3} \times {10}^{4 + 2} = 3.0 \times {10}^{6}}$