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Chapter 5: Problem 123

Explain the negative exponent rule and give an example.

Short Answer

Expert verified
The rule of negative exponents states that for any nonzero number a and integer n, \(a^{-n} = \frac{1}{a^n}\). It means a negative exponent 'flips' the base. For example, \(4^{-2} = \frac{1}{4^{2}} = \frac{1}{16}\).

Step by step solution

01

Definition of Negative Exponential Rule

The rule of negative exponents states that when a base is raised to a negative exponent, it is equivalent to 1 divided by the base raised to its absolute exponent. In mathematical terms, for any nonzero number a and integer n, \(a^{-n} = \frac{1}{a^n}\).
02

Explanation of the Negative Exponential Rule

In simple terms, a negative exponent indicates reciprocal or 'flipping' of the base. The reason is that, according to the laws of exponents, subtracting in the exponent is equivalent to division in the base. Therefore, when you have a negative exponent, it means you have subtracted more than you have, and you need to 'pay back', which you do by flipping the base and changing the exponent to positive.
03

Example of the Negative Exponential Rule

Let's consider an example with the base as 4 and the exponent as -2, i.e., \(4^{-2}\). According to the negative exponent rule, this equals \( \frac{1}{4^{2}} \) . Simplifying further, it equals \( \frac{1}{16} \).

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Most popular questions from this chapter

The sum, \(S_{n}\), of the first n terms of an arithmetic sequence is given by $$ S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right), $$ in which \(a_{1}\) is the first term and \(a_{n}\) is the nth term. The sum, \(S_{n}\), of the first \(n\) terms of a geometric sequence is given by $$ S_{n}=\frac{a_{1}\left(1-r^{n}\right)}{1-r}, $$ in which \(a_{1}\) is the first term and \(r\) is the common ratio \((r \neq 1)\). Determine whether each sequence is arithmetic or geometric. Then use the appropriate formula to find \(S_{10}\), the sum of the first ten terms. \(3,-6,12,-24, \ldots\)

Express each number in scientific notation. 32,000

Use the distributive property to simplify the radical expressions \(\sqrt{3}(5+\sqrt{3})\)

You will develop geometric sequences that model the population growth for California and Texas, the two most populated U.S. states. The table shows the population of Texas for 2000 and 2010 , with estimates given by the U.S. Census Bureau for 2001 through \(2009 .\) $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \text { Year } & \mathbf{2 0 0 0} & \mathbf{2 0 0 1} & \mathbf{2 0 0 2} & \mathbf{2 0 0 3} & \mathbf{2 0 0 4} & \mathbf{2 0 0 5} \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 20.85 & 21.27 & 21.70 & 22.13 & 22.57 & 23.02 \\ \hline \end{array} $$ $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Year } & \mathbf{2 0 0 6} & \mathbf{2 0 0 7} & \mathbf{2 0 0 8} & \mathbf{2 0 0 9} & \mathbf{2 0 1 0} \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 23.48 & 23.95 & 24.43 & 24.92 & 25.15 \\ \hline \end{array} $$ a. Divide the population for each year by the population in the preceding year. Round to two decimal places and show that Texas has a population increase that is approximately geometric. b. Write the general term of the geometric sequence modeling Texas's population, in millions, \(n\) years after \(1999 .\) c. Use your model from part (b) to project Texas's population, in millions, for the year 2020 . Round to two decimal places.

Use rules of divisibility to determine whether each number given is divisible by a. 2 b. 3 c. 4 d. 5 e. 6 f. 8 g. 9 h. 10 i. 12 . \(6,126,120\)
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