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Chapter 4: Problem 13

Evaluate each expression. Do not use a calculator. $$16^{-3 / 4}$$

Short Answer

Expert verified
The value of \( 16^{-3/4} \) is \( \frac{1}{8} \).

Step by step solution

01

Identify the Base and Exponent

The expression given is \( 16^{-3/4} \). Here, 16 is the base, and \(-3/4\) is the exponent. The negative exponent indicates that the expression is the reciprocal of the base raised to a positive exponent.
02

Rewriting the Negative Exponent

Rewrite the expression using the negative exponent rule: \( a^{-n} = \frac{1}{a^n} \). Thus, \( 16^{-3/4} = \frac{1}{16^{3/4}} \).
03

Simplify the Expression with Positive Exponent

To simplify \( 16^{3/4} \), use the property that \( a^{m/n} = \sqrt[n]{a^m} \). Here, we have \( 16^{3/4} = (16)^{1/4}^3 \).
04

Evaluate the Fourth Root

Evaluate \( 16^{1/4} \). The fourth root of 16 is the number which raised to the power of 4 equals 16. Since \( 2^4 = 16 \), \( 16^{1/4} = 2 \).
05

Raise to the Power of 3

Now, raise the result from Step 4 to the power of 3: \( 2^3 = 8 \).
06

Determine the Reciprocal

From Step 2, since our exponent is originally negative, take the reciprocal of the result we found: \( \frac{1}{8} \).

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Key Concepts

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

Negative Exponents
In mathematics, we often encounter expressions with negative exponents. A negative exponent signifies that the base should be inverted or turned into its reciprocal, which means flipping it to the bottom of a fraction if it is initially on the top, and vice versa. For instance:
  • If you have an expression such as \( a^{-n} \), it can be rewritten as \( \frac{1}{a^n} \).
  • This principle helps simplify expressions by allowing us to switch between negative and positive exponents, which often makes calculations more manageable.
Therefore, when you see an expression like \( 16^{-3/4} \), initially consider it as \( \frac{1}{16^{3/4}} \). This transformation is the crucial step that simplifies the process, as it converts a more complex operation into something that's easier to evaluate.
Rational Exponents
Rational exponents are a type of exponent that involve fractions. Specifically, they describe roots along with the usual power of a number and provide a way to express radicals more compactly. Here's what you need to know:
  • When you see an expression like \( a^{m/n} \), it implies two operations. The base \( a \) is first raised to the power of \( m \), and then you take the \( n^{th} \) root of that result, or vice versa.
  • This is interpreted as \( \sqrt[n]{a^m} \). You could also see it as doing the nth root first, then raising it to the m-th power.
In our exercise, \( 16^{3/4} \) stands for taking the fourth root of 16 and then cubing the outcome. This dual operation makes rational exponents a versatile tool for simplifying and evaluating expressions without using a calculator.
Roots
Roots are fundamental operations in mathematics, especially when handling rational exponents. They essentially reverse the process of exponentiation.
  • The nth root of a number \( a \), written as \( \sqrt[n]{a} \), is the value that, when raised to the nth power, equals \( a \).
  • For example, the square root (\( \sqrt{} \)) is the most common root operation, but other roots like the cube root \( \sqrt[3]{} \) and the fourth root \( \sqrt[4]{} \) are just as important in various applications.
In the given problem, we identify \( 16^{1/4} \) as the fourth root of 16, which equals 2, since 2 raised to the fourth power gives us 16. Understanding roots allows you to easily decompose complex expressions and is essential for manipulating equations with rational exponents.

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Most popular questions from this chapter

In Exercises \(109-116\), describe the graph of the equation as either a circle or a parabola with a horizontal axis of symmetry. Then, determine two functions, designated by \(y_{1}\) and \(y_{2},\) such that their union will give the graph of the given equation. Finally, graph \(y_{1}\) and \(y_{2}\) in the given viewing window. $$\begin{aligned} &x^{2}+y^{2}=81\\\ &[-15,15] \text { by }[-10,10] \end{aligned}$$

Solve each problem. Volume of a Gas Natural gas provides \(25 \%\) of U.S. energy. The volume of a gas varies inversely with the pressure and directly with the temperature. (Temperature must be measured in degrees Kelvin (K), a unit of measurement used in physics.) If a certain gas occupies a volume of 1.3 liters at \(300 \mathrm{K}\) and a pressure of 18 newtons per square centimeter, find the volume at \(340 \mathrm{K}\) and a pressure of 24 newtons per square centimeter.

Animal Pulse Rate and Weight According to one model, the rate at which an animal's heart beats varies with its weight. Smaller animals tend to have faster pulses, whereas larger animals tend to have slower pulses. The table lists average pulse rates in beats per minute (bpm) for animals with various weights in pounds (lb). Use regression (or some other method) to find values for \(a\) and \(b\) so that \(f(x)=a x^{b}\) models these data. $$\begin{array}{|l|r|r|r|r|r|} \hline \text { Weight (in Ib) } & 40 & 150 & 400 & 1000 & 2000 \\ \hline \text { Pulse (in bpm) } & 140 & 72 & 44 & 28 & 20 \end{array}$$ (IMAGE CANT COPY)

Consider the expression \(16^{-3 / 4}\) (a) Simplify this expression without using a calculator. Give the answer in both decimal and \(\frac{a}{b}\) form. (b) Write two different radical expressions that are equivalent to it, and use your calculator to evaluate them to show that the result is the same as the decimal form you found in part (a). (c) If your calculator has the capability to convert decimal numbers to fractions, use it to verify your results in part (a).

Solve each problem. If \(y\) varies directly with \(x\) and inversely with \(m^{2}\) and \(r^{2},\) and \(y=\frac{5}{3}\) when \(x=1, m=2,\) and \(r=3,\) find \(y\) if \(x=3\) \(m=1,\) and \(r=8.\)
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