Exponents and Radicals
Using Integers as Exponents
Definition 5.1 Exponent of Zero
If \(b\) is a nonzero real number, then
$$ b^0 = 1 $$Definition 5.2 Negative Exponent
If \(n\) is a positive integer and \(b\) is a nonzero real number, then
$$ b^{-n} = \frac{1}{b^n} $$Property 5.2 Properties of Integer Exponents
If \(m\) and \(n\) are integers and \(a\) and \(b\) are real numbers, (and \(b \neq 0\) whenever it appears in a denominator), then
- $$ b^n \cdot b^m = b^{n + m}$$
- $$ (b^n)^m = b^{nm} $$
- $$ (ab)^n = a^n b^n $$
- $$ \left(\frac{a}{b} \right)^n = \frac{a^n}{b^n} $$
- $$ \frac{b^n}{b^m} = b^{n - m} $$
Roots and Radicals
The symbol \(\sqrt{}\) called a radical sign, is used to designate the nonnegative or principal square root. The number under the radical sign is called the radicand. The entire expression, such as \(\sqrt{16}\), is called a radical.
Definition 5.3 Principal Square Root
If \(a \leq 0\) and \(b \leq 0\), then \(\sqrt{b} = a\) if and only if \(a^2 = b\); \(a\) is called the principal square root of \(b\).
Definition 5.5 \(n\)th Root of a Number
The \(n\)th root of \(b\) is \(a\) if and only if \(a^n = b\).
We can make the following generalizations:
If \(n\) is an even positive integer
- Every positive real number has exactly two real nth roots, one positive and one negative.
- Negative real numbers do not have real nth roots
If \(n\) is an odd positive integer greater than \(1\)
- Every real number has exactly one real nth root.
- The real nth root of a positive number is positive.
- The real nth root of a negative number is negative.
The symbol \(\sqrt[n]{}\) designates the principal \(n\)th root.
Property 5.3
- \((\sqrt[n]{b})^n = b\), \(n\) is any positive integer greater than \(1\).
- \(\sqrt[n]{b^n} = b\), \(n\) is any positive integer greater than \(1\) if \(b \geq 0 \); \(n\) is an odd positive integer greater than \(1\) if \(b < 0 \)
Property 5.4
when \(\sqrt[n]{b}\) and \(\sqrt[n]{c}\) are real numbers.
Expressing a Radical in Simplest Radical Form
Property 5.5
when \(\sqrt[n]{b}\) and \(\sqrt[n]{c}\) are real numbers and \(c \neq 0\)
A radical is said to be in simplest radical form if the following conditions are satisfied.
- No fraction appears with a radical sign.
- No radical appears in the denominator.
- No radicand, when expressed in prime-factored form, contains a factor raised to a power equal to or greater than the index.
Combining Radicals and Simplifying Radicals That Contain Variables
Note that in order to be added or subtracted, radicals must have the same index and the same radicand.
Products and Quotients Involving Radicals
The factors \(a + b\) and \(a - b\) are called conjugates.
Equations Involving Radicals
We often refer to equations that contain radicals with variables in a radicand as radical equations.
Property 5.6
Let \(a\) and \(b\) be real number and \(n\) be a positive integer
If \(a = b\) then \(a^n\) = \(b^n\)
In general, raising both sides of an equation to a positive integral power produces an equation that has all of the solutions of the original equation, but it may also have some extra solutions that do not satisfy the original equation. Such extra solutions are called extraneous solutions. Therefore, when using Property 5.6, you must check each potential solution in the original equation.
Merging Exponents and Roots
Definition 5.6
If \(b\) is a real number, \(n\) is a positive integer greater than \(1\) and \(\sqrt[n]{b}\) exists then
$$ b^{\frac{1}{n}} = \sqrt[n]{b} $$Definition 5.7
If \(\frac{m}{n}\) is a rational number, where \(n\) is a positive integer greater than \(1\) and \(b\) is a real number such that \(\sqrt[n]{b}\) exists then
$$ b^{\frac{m}{n}} = (\sqrt[n]{b})^m $$Multiplying and Dividing Radicals with Different Indexes
The link between exponents and roots also provides a basis for multiplying and dividing some radicals even if they have different indexes. The general procedure is as follows:
- Change from radical form to exponential form.
- Apply the properties of exponents.
- Then change back to radical form.