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Squares and other powers

An exponent, or a power, is mathematical shorthand for repeated multiplications. For example, the exponent “2” means to multiply the base for that exponent by itself (in the example here, the base is “5”):

5 2 = 5 × 5 = 25

The exponent is “2” and the base is the number “5.” This expression (multiplying a number by itself) is also called a square. Any number raised to the power of 2 is being squared. Any number raised to the power of 3 is being cubed:

5 3 = 5 × 5 × 5 = 125

A number raised to the fourth power is equal to that number multiplied by itself four times, and so on for higher powers. In general:

n x = n × n x 1

Calculating percents

A percent is a way of expressing a fractional amount of something using a whole divided into 100 parts. A percent is a ratio whose denominator is 100. We use the percent symbol, %, to show percent. Thus, 25% means a ratio of 25 100 , 3% means a ratio of 3 100 , and 100 % percent means 100 100 , or a whole.

Converting percents

A percent can be converted to a fraction by writing the value of the percent as a fraction with a denominator of 100 and simplifying the fraction if possible.

25% = 25 100 = 1 4

A percent can be converted to a decimal by writing the value of the percent as a fraction with a denominator of 100 and dividing the numerator by the denominator.

10% = 10 100 = 0.10

To convert a decimal to a percent, write the decimal as a fraction. If the denominator of the fraction is not 100, convert it to a fraction with a denominator of 100, and then write the fraction as a percent.

0.833 = 833 1000 = 83.3 100 = 83.3%

To convert a fraction to a percent, first convert the fraction to a decimal, and then convert the decimal to a percent.

3 4 = 0.75 = 75 100 = 75%

Suppose a researcher finds that 15 out of 23 students in a class are carriers of Neisseria meningitides . What percentage of students are carriers? To find this value, first express the numbers as a fraction.

carriers total students = 15 23

Then divide the numerator by the denominator.

15 23 = 15 ÷ 23 0.65

Finally, to convert a decimal to a percent, multiply by 100.

0.65 × 100 = 65%

The percent of students who are carriers is 65%.

You might also get data on occurrence and non-occurrence; for example, in a sample of students, 9 tested positive for Toxoplasma antibodies, while 28 tested negative. What is the percentage of seropositive students? The first step is to determine the “whole,” of which the positive students are a part. To do this, sum the positive and negative tests.

positive + negative = 9 + 28 = 37

The whole sample consisted of 37 students. The fraction of positives is:

positive total students = 9 37

To find the percent of students who are carriers, divide the numerator by the denominator and multiply by 100.

9 37 = 9 ÷ 37 0.24 0.24 × 100 = 24%

The percent of positive students is about 24%.

Another way to think about calculating a percent is to set up equivalent fractions, one of which is a fraction with 100 as the denominator, and cross-multiply. The previous example would be expressed as:

9 37 = x 100

Now, cross multiply and solve for the unknown:

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Source:  OpenStax, Microbiology. OpenStax CNX. Nov 01, 2016 Download for free at http://cnx.org/content/col12087/1.4
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