2.1: Powers and Roots
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 \(1.44\, \sqrt{\quad }\)