2.1: Powers and Roots
- Page ID
- 20032
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)As we learned in Unit 1, we can use the following operations on real numbers: addition, subtraction, multiplication, division, and absolute value. In addition to these operations, we can also apply powers and roots.
Powers
Powers are as the notation for multiplying a number by itself multiple times. Another name for a power is an exponent. The power or exponent is denoted as a superscript.
\[a^n=\underbrace{(a\cdot a\cdot a\cdot \cdots \cdot a)}_{n\ times}\]
In this notation, the a is called the “base” and the \(n\) is called the power (or exponent).
Examples:
- \(5^3=5\cdot 5\cdot 5=125\)
- \({(-2)}^4=(-2)(-2)(-2)(-2)=16\)
- \({(2.3)}^2=(2.3)(2.3)=5.29\)
- \({(2/3)}^5=2/3\cdot 2/3\cdot 2/3\cdot 2/3\cdot 2/3=32/243\)
Roots
Roots are obtained by “undoing” a power. There are many types of roots, we will look at square roots, cube roots, and fourth roots.
- The square root of a number, \(a\), is \(b\) if\(\ b^2=a\)
- The cube root of a number, \(a\), is \(b\) if \(b^3=a\)
- The fourth root of a number, \(a\), is \(b\) if \(b^4=a\)
- We can generalize this to say the n\({}^{th}\) root of a number, a, is b if \(b^n=a\)
We use a radical symbol to represent the operation of a root: \(\sqrt{\quad }\)
- Square root symbol: \(\sqrt{\quad }\)
- Cube root symbol: \(\sqrt[3]{\quad}\)
- Fourth root symbol: \(\sqrt[4]{\quad}\)
- n\({}^{th}\) root symbol: \(\sqrt[n]{\quad}\)
The number under a radical symbol is called the radicand.
Examples:
- The square root of 9 is denoted as \(\sqrt{9}\), since \(3\wedge 2=9,\ then\ \sqrt{9}=3\).
- \(\sqrt{49}\), since \(7^2=49,\ then\ \sqrt{49}=7\).
- The cube root of 8 is denoted as \(\sqrt{}\), since \(2^3=8,\ then\ \sqrt{8}=2\).
- \(\sqrt{1.44}=1.2\) since \({\left(1.2\right)}^2=1.44\)
- \(\sqrt{\frac{36}{121}}=6/11\) since\({\left(\frac{6}{11}\right)}^2=36/121\)
To compute a root:
- On a scientific display calculator, type the radical symbol first, then the radicand followed by enter.
Example: \(\sqrt{1.44}\) the keystrokes would be \(\sqrt{\quad }\ 1.44\, Enter\)
- On a non-display calculator, type the radicand first, then the radical symbol (do not press =)
Example: \(\sqrt{1.44}\) the keystrokes would be \(1.44\, \sqrt{\quad }\)