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Chapter 4: Problem 16

Determine the number that can be used in place of the question mark to make the equation true. $$\left(\frac{1}{5}\right)^{?}=\frac{1}{125}$$

Short Answer

Expert verified
? = 3

Step by step solution

01

- Recognize the base

Identify the base of the expression on the left-hand side. The base in the left-hand side is \(\frac{1}{5}\).
02

- Rewrite the right-hand side

Express the right-hand side of the equation in terms of the same base. \(\frac{1}{125}\) can be written as \(\left( \frac{1}{5} \right)^{3} \).
03

- Equate the exponents

Since the bases are now the same, you can set the exponents equal to each other: \( ? = 3 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Functions
An exponential function is a mathematical expression where a constant base is raised to a variable exponent. It looks like this: \(a^x\). Here, 'a' is the base and 'x' is the exponent.
Exponential functions grow or shrink rapidly. They are used in many fields such as science, finance, and population modeling.
Examples of exponential functions include:
  • Growth: \(2^x\)
  • Decay: \(\frac{1}{3}^x\)
In the given exercise, the exponential function on the left-hand side is \((\frac{1}{5})^?\). This shows exponential decay because the base \(\frac{1}{5}\) is a fraction less than 1.
Base Recognition
Base recognition means identifying the base in an exponential function. Recognizing the base is the first step to solving exponential equations.
To solve \((\frac{1}{5})^? = \frac{1}{125}\), we need to know that the base on the left-hand side is \(\frac{1}{5}\).
Next, we must express the right-hand side in terms of the same base. Notice that \(125 = 5^3\). Therefore, \(\frac{1}{125} = (\frac{1}{5})^3\).
When both sides of the equation have the same base, it becomes easier to solve for the unknown exponent.
Equating Exponents
After rewriting both sides with the same base, you can equate the exponents. This is because the bases are already matched.
If we have \((\frac{1}{5})^x = (\frac{1}{5})^3\), then we can simply say that \(x = 3\).
This is known as 'equating exponents'. This step transforms the original exponential equation into a simple linear equation.
Here are the steps we followed:
  • Recognize the base.
  • Rewrite each side with the same base.
  • Equate the exponents to solve for the variable.
This method is efficient and widely used for solving exponential equations.

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Most popular questions from this chapter

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Use the following definition. In chemistry, the \(\mathrm{pH}\) of a solution is defined to be $$\mathrm{pH}=-\log \left[H^{+}\right],$$ where \(H^{+}\) is the hydrogen ion concentration of the solution in moles per liter. Distilled water has a pH of approximately 7. A substance with a pH under 7 is called an acid, and one with a pH over 7 is called a base. The hydrogen ion concentration of orange juice is \(10^{-3.7}\) moles per liter. Find the pH of orange juice.

Computers per Capita The number of personal computers per 1000 people in the United States from 1990 through 2010 is given in the accompanying table (Consumer Industry Almanac, www.c-i-a.com). a. Use exponential regression on a graphing calculator to find the best- fitting curve of the form \(y=a \cdot b^{x},\) where \(x=0\) corresponds to 1990. b. Write your equation in the form \(y=a e^{e x}.\) c. Assuming that the number of computers per 1000 people is growing continuously, what is the annual percentage rate? d. In what year will the number of computers per 1000 people reach \(1500 ?\) e. Judging from the graph of the data and the curve, does the exponential model look like a good model? $$\begin{array}{|l|c|} \hline \text { Year } & \begin{array}{c} \text { Computers } \\ \text { per 1000 } \end{array} \\ \hline 1990 & 192 \\ 1995 & 321 \\ 2000 & 628 \\ 2005 & 778 \\ 2010 & 932 \\ \hline \end{array}$$
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