# Question 78c81

May 18, 2017

$5.000 \times {10}^{-} 1$

#### Explanation:

Scientific notation is written as a number multiplied by a power of $10.$ The number must have ONE digit before the decimal point.

Let's look at it using decimals, fractions and indices.

$0.5 = \frac{5}{10} = 5 \times {10}^{-} 1$

($\frac{1}{x} ^ 2 = {x}^{-} 2 , \text{ } \therefore \frac{1}{10} = {10}^{-} 1$)

However, we need to keep the same number of significant figures in the answer as we started with, so the zeroes need to be shown as well.

If there are more place holders than in this example, then you count how many places the decimal point has moved.

$2 \textcolor{b l u e}{36 , 000 , 000} . = 2.36 \times {10}^{\textcolor{b l u e}{8}}$
The number is big, therefore the index of 10 is positive.

0color(red)(.000,000,000,6)41 = 6.41 xx 10^color(red)(-10#
The number is a very small decimal therefore the index of 1 is negative

May 18, 2017

$5.000 \times {10}^{-} 1$

#### Explanation:

The answer in scientific notation must be written as one digit to the left of the decimal and the rest of the digits that are significant to the right of the decimal multiplied by a power of ten.

$0.5 = \frac{5}{10}$

dividing by 10 is the same as ${10}^{-} 1$ so

$5. \times {10}^{-} 1 = 0.5$

The zeros in $.5000$ have no mathematical value as place holders. The best reason for writing the zeros is that the zero's represent a measured value. significant digits represent numbers that have been measured. It is possible the zero's are simply a result of a division or multiplication operation and have not been measured. However in writing in scientific notation is best to assume that they have been measured. Therefore the zero's should be written as part of the number in scientific notation. so

$0.5000 = 5.000 \times {10}^{-} 1$ in scientific notation.