hour of growth, there are two bacteria (Recall: 2 = 2 1 ) • the point (7, 128) farther down the table means that after seven hours since the experiment began, there are 128 bacteria (Recall: 128 = 2 7 ) Example Using the equation for doubling, predict the number of bacteria that will have grown after 20 hours. Solution The equation for doubling refers to y = 2 x . If the same rate of growth is maintained for 20 hours, then the population would reach the following size: y = 2 20 y = 1 048 576 The answer, 1 048 576, was determined by evaluating the expression 2 20 on a calculator. Although it seems precise, for practical purposes, it should be taken as approximately one million bacteria. Exponent Law for Division Doubling is a word used to describe the rate at which the numbers 2, 4, 8, 16, and so on are increasing from term to term in the y = 2 x column. Other patterns that exist in this same table are so important that they are considered to be mathematical laws, known as exponent laws. To investigate them, look at the table of y = 2 x in the previous section and proceed as follows: • Take any two numbers from the y = 2 x column, for example, 1024 and 128.
• Divide the larger number, 1024, by the smaller number, 128: 1024 = 8 128 • Find the result, 8, also in the y -column of the table. Now, rewrite the equation above in exponential form: 1024 = 2 10 = 2 3 128 2 7 Look at the exponents in the rewritten equation. Do you see a pattern? 10 – 7 = 3. Choose two more numbers from the table and try the same thing. Does the pattern repeat itself? A more general rule can be stated for exponential expressions, as long as they have the same base within the expression; it does not have to be base 2. The general rule is, a m = a m – n a n , where the base, a , stands for any number. This is known as the exponent law for division. Expressed in words, it says that when dividing numbers in the form of powers with the same base, you can form a new expression, with the same base, whose exponent is found by subtracting the exponent of the denominator from the exponent of the numerator. Example Apply the exponent law or division to simplify the expression 2 8 . 2 1 Solution 2 8 = 2 8 − 1 = 2 7 2 1 Exponent Law for Multiplication The same pattern you have been studying could also be stated as a multiplication rule, to go along with the division rule. For example, • a calculator would show that 8 × 16 = 128. • converting each of the factors in this equation to a power, using the previous table for y = 2 x , you find that 2 3 × 2 4 = 2 7 . Such patterns in the table for y = 2 x , and then for other simple powers of the form y = a x , eventually lead to the exponent law for multiplication: a m × a n = a m+n . Make sure that you have made the connections between the patterns and the exponent rules before you read on.
Example Use the exponent law for multiplication to find values for the following and locate the answers in the table for y = 2 x , if possible: a) Simplify 2 3 × 2 2 .
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- Spring '17
- Math, Exponential Function, Exponentiation, Power of two