# Question e817a

Sep 12, 2016

$2.0000 \cdot {10}^{3}$

#### Explanation:

As you know, a number written in scientific notation will have the form

$\textcolor{w h i t e}{a a} \textcolor{b l u e}{m} \times {10}^{\textcolor{p u r p \le}{n} \textcolor{w h i t e}{a} \stackrel{\textcolor{w h i t e}{a a a a a a}}{\leftarrow}} \textcolor{w h i t e}{a \textcolor{b l a c k}{\text{the")acolor(purple)("exponent}} a a}$
$\textcolor{w h i t e}{\frac{a}{a} \textcolor{b l a c k}{\uparrow} a a a a}$
$\textcolor{w h i t e}{\textcolor{b l a c k}{\text{the")acolor(blue)("mantissa}} a}$

In the vast majority of cases, you will be dealing with normalized scientific notation, which must have

1 <= |m| < 10" " " "color(orange)("(*)")#

Now, the thing to remember about scientific notation and significant figures is that the mantissa must always have the same number of sig figs as the number written in standard notation.

In your case, the number $2000.0$ has five sig figs, which means that the mantissa must also have five sig figs.

All you have to do now is convert $2000.0$ to a number that satisfies condition $\textcolor{\mathmr{and} a n \ge}{\text{(*)}}$ and has five sig figs.

As you know, zeros that follow a decimal point are significant if they also follow a non-zero digit. In this case, you can write $2000.0$ as

$2.0000 \to$ one non-zero digit and four significant zeros for a total of five sig figs

This means that the scientific notation for $2000.0$ will be

$2000.0 = 2.0000 \cdot {10}^{3}$

The exponent is positive because you're moving the decimal point $3$ places to the left.