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Chapter 1: Problem 38

Evaluate each expression without using a calculator. $$ 16^{-3 / 4} $$

Short Answer

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

Step by step solution

01

Understand Exponent Notation

The expression \(16^{-\frac{3}{4}}\) is composed of a base, 16, and an exponent, -3/4. A negative exponent indicates that the base, 16, should be taken as the reciprocal raised to the positive exponent, \/\frac{3}{4}.
02

Rewrite with Positive Exponent

Convert the expression with the negative exponent to a positive one. \[16^{-\frac{3}{4}} = \frac{1}{16^{\frac{3}{4}}}\]
03

Simplify the Exponentiation

Now, find \(16^{\frac{3}{4}}\). Start by understanding that \(16^{\frac{3}{4}}\) means taking the 4th root of 16 first and then cubing the result. \[16^{\frac{1}{4}} = \sqrt[4]{16} = 2\] Thus, \(16^{\frac{3}{4}}\) becomes \(2^3\).
04

Evaluate the Power

Calculate \(2^3\) which is the cube of 2. \[2^3 = 2 \times 2 \times 2 = 8\]
05

Determine the Reciprocal

Substitute \(16^{\frac{3}{4}}\) back into the reciprocal expression to get the final evaluation. \[\frac{1}{16^{\frac{3}{4}}} = \frac{1}{8}\].

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Key Concepts

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

Negative Exponents
When we see a negative exponent, it means we are looking at the reciprocal of the base raised to the opposite positive exponent. This might sound a bit confusing, so let's clarify with examples:
  • For example, if you see \(a^{-n}\), it should be interpreted as \(\frac{1}{a^n}\).
  • This concept helps in transforming complex exponential problems into simpler fractions.
With a clear understanding of negative exponents, solving equations like \(16^{-\frac{3}{4}}\) becomes easier, as we immediately know to rewrite it as \(\frac{1}{16^{\frac{3}{4}}}\). It simplifies the problems allowing easy manipulation of more complex calculations.
Fractional Exponents
Fractional exponents indicate both roots and powers. For example, the expression \(a^{m/n}\) means take the \(n\)-th root of \(a\) and then raise the result to the \(m\)-th power.
  • Think of it like doing two operations. First, extract the root, and next, raise it to the given power.
  • In our example of \(16^{-\frac{3}{4}}\), once simplified, it involves finding the fourth root of 16, which is 2.
Then, you raise 2 to the third power, resulting in 8. This concept is fundamental as it connects roots and exponents in the same expression, providing a shortcut to solving expressions without breaking down each part separately.
Simplifying Radicals
Simplifying radicals involves breaking down expressions to their simplest radical form. Consider the expression \(\sqrt[4]{16}\). Here's how you approach it:
  • Look for a number that can multiply itself to equal the base inside the radical, in this case, four times over, since it's a fourth root.
  • For \(16\), the fourth root is 2 because \(2 \times 2 \times 2 \times 2 = 16\).
Once you find the simplified expression, it becomes much easier to handle further calculations, such as raising this to another power. Understanding these intermediate steps can demystify complicated-looking expressions and give you a methodical way to handle them.

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Most popular questions from this chapter

For each pair of functions \(f(x)\) and \(g(x)\), find and fully simplify a. \(f(g(x))\) and b. \(g(f(x))\) $$ f(x)=x^{3}+1 ; \quad g(x)=\sqrt[3]{x-1} $$

For each function, find and simplify \(\frac{f(x+h)-f(x)}{h}\). (Assume \(h \neq 0 .\) ) $$ \begin{array}{l} f(x)=x^{3} \\ \text { [Hint: Use } \left.(x+h)^{3}=x^{3}+3 x^{2} h+3 x h^{2}+h^{3} .\right] \end{array} $$

For the functions \(y_{1}=\left(\frac{1}{3}\right)^{x}, y_{2}=\left(\frac{1}{2}\right)^{x}\), \(y_{3}=2^{x}\), and \(y_{4}=3^{x}\) a. Predict which curve will be the highest for large values of \(x\). b. Predict which curve will be the lowest for large values of \(x\). c. Check your predictions by graphing the functions on the window \([-3,3]\) by \([0,5]\). d. From your graph, what is the common \(y\) -intercept? Why do all such exponential functions meet at this point?

For each function, find and simplify \(\frac{f(x+h)-f(x)}{h}\). (Assume \(h \neq 0 .\) ) $$ f(x)=\frac{2}{x} $$

Find the slope (if it is defined) of the line determined by each pair of points. \((3,-1)\) and \((5,7)\)
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