10: Radicals
- Page ID
- 45080
By the end of this chapter, the student should be able to
- Simplify radical expressions
- Rationalize denominators (monomial and binomial) of radical expressions
- Add, subtract, and multiply radical expressions with and without variables
- Solve equations containing radicals and radical functions
- Solve equations containing rational exponents
Radicals are a common concept in algebra. In fact, we think of radicals as reversing the operation of an exponent. Hence, instead of the “square” of a number, we take the “square root” a number; instead of the “cube” of a number, we take the “cube root” a number, and so on. Square roots are the most common type of radical used in algebra.
The radical sign, when first used, was an R with a line through the tail, ℞, similar to our medical prescription symbol. The R came from the latin, “radix”, which can be translated as “source” or “foundation.” It wasn’t until the 1500s that our current symbol was first used in Germany, but even then it was just a check mark with no bar over the numbers, √.
If \(a\) is a positive real number, then the principal square root of a number \(a\) is defined as \[\sqrt{a}=b\text{ if and only if }a=b^2,\nonumber\] where \(b > 0\). The \(\sqrt{\quad}\) is the radical symbol, and \(a\) is called the radicand.
If given something like \(\sqrt[3]{a}\), then \(3\) is called the root or index; hence, \(\sqrt[3]{a}\) is called the cube root or third root of \(a\). In general, \[\sqrt[n]{a}=b\text{ if and only if }a = b^n\nonumber\] If \(n\) is even, then \(a\) and \(b\) must be greater than or equal to zero. If \(n\) is odd, then \(a\) and \(b\) can be any real number.
Here are some examples of square roots:
\[\begin{array}{ll}\sqrt{1}=1 &\sqrt{121}=11 \\ \sqrt{4}=2 &\sqrt{625}=25 \\ \sqrt{9}=3& \sqrt{-81}=\text{ not a real number}\end{array}\nonumber\]
The final example, \(\sqrt{−81}\) is not a real number. There is a future section in which will discuss examples like \(\sqrt{-81}\). Recall, if the root is even, then the radicand must be greater than or equal to zero and since \(−81 < 0\), then there is no real number in which we can square and will result in \(−81\), i.e., \(?^2 = −81\). So, for now, when we obtain a radicand that is negative and the root is even, we say that this number is not a real number. There is a type of number where we can evaluate these numbers, but just not a real one.
- 10.4: Rationalize Denominators
- When given a quotient with radicals, it is common practice to leave an expression without a radical in the denominator. After simplifying an expression, if there is a radical in the denominator, we will rationalize it so that the denominator is left without any radicals. We start by rationalizing denominators with square roots, and then extend this idea to higher roots.
- 10.5: Radicals with Mixed Indices
- Knowing that a radical has the same properties as exponents (written as a ratio) allows us to manipulate radicals in new ways. One thing we are allowed to do is reduce, not just the radicand, but the index as well.