1.
\(2^5\)
Section 5.3 Roots
In Section 5.2 we solved expressions with quadratics using the square root. Here we will briefly describe roots in general including notation and their rate.
Subsection 5.3.1 Definition of Roots
As implied by their use in solving square roots are an opposite concept to squares. For example \(3^2=9\) means that 9 is the result of multiplying 3 by itself. \(\sqrt{16}=4\) means that 4 is a number that multiplied by itself is 16. This means in general
\begin{equation*}
\sqrt{n^2}=n.\text{.}
\end{equation*}
There is one detail that we will not use, but should be acknowledged. Note that \((3)^2=9\) and \((-3)^2=9\text{.}\) Thus \(\sqrt{9}\) might be considered to have two solutions. We saw this in Subsubsection 5.2.1.3. When solving using inversion as in Subsubsection 5.2.1.1 we will ignore the negative solution which will not be useful in the problems asked.
Subsection 5.3.2 Generalized Roots
Subsubsection 5.3.2.1 Generalized Powers
Just as we can multiply a number by itself (e.g., \(3^2=3 \cdot 3 = 9\)) we can multiply a number by itself more than once (e.g., \(3^3=3 \cdot 3 \cdot 3 = 27\)). In general
\begin{equation*}
3^m = \overbrace{3 \cdot 3 \cdot \ldots 3}^m.
\end{equation*}
Reading Questions Reading Questions
Evaluate each of these by multiplying enough times
2.
\(4^3\)
3.
\((1.9)^4\)
Subsubsection 5.3.2.2 Generalized Roots
Just as there are square roots to solve problems involving squares, there are roots for other powers as well. These are denoted with a small number to show which root. For example this is a third root:
\begin{equation*}
\sqrt[3]{8}=2.
\end{equation*}
Just as with square roots we can use a device to calculate the values. The device may have a key like \(\sqrt[m]{x}\text{.}\)
Another notation for roots is a type of exponent. For example,
\begin{equation*}
\sqrt{9}=9^{1/2}=3.
\end{equation*}
Likewise,
\begin{equation*}
\sqrt[3]{8}=8^{1/3}=2.
\end{equation*}
This notation can be used when solving problems with powers.
Reading Questions Reading Questions
Evaluate each of these using a device.
1.
\(\sqrt[3]{27}\)
2.
\(\sqrt[3]{64}\)
3.
\(\sqrt[5]{32}\)
4.
\(\sqrt[4]{20}\)
Subsection 5.3.3 Rates of Roots
We have compared rates of linear, quadratic, and exponential data (Section 3.2). Here we learn about rates by considering the rate of roots.
To compare rates see Figure 5.3.1. Notice that the rate is eventuallymuch slower than even a linear. Because a linear is slower than a linear it is also slower than a quadratic or exponential.
With rates it is important to test if a slow growth rate is leveling off or not. We will show that a square root (and other roots) do not level off. Note in Table 5.3.2 we can produce an arbitrarily large result from a square root. If it leveled off at some number like 2000, we would not have a result larger, but \(\sqrt{3000^2}=3000\text{.}\) We can do this for any possible height for leveling off.
| \(n\) | \(\sqrt{n}\) |
| \(1^2=1\) | \(1\) |
| \(2^2=4\) | \(2\) |
| \(3^2=9\) | \(3\) |
| \(10^2=100\) | \(10\) |
| \(n^2\) | \(n\) |
