Square Roots and Other Radicals
In this section, we will review some facts about square roots and other radicals which are relevant in calculus.
Here is the formal definition of a square root.
That is, x is the non-negative number whose square is a. For example, since (0.3)^{2} = 0.09.
The expression is defined only when a >= 0, and so an expression like makes sense only if 2-3x >= 0, or, solving the inequality, if x <= 2/3. When a function is defined using a square root, finding its domain often involves solving an inequality.
So to find the domain of a function with a quadratic expression under the root sign (that is, the radicand is quadratic), one might have to solve a quadratic inequality. Example.
Properties of square roots
Square roots have the following properties.
where a, b >= 0 | ||
where a >= 0 and b > 0 | ||
for any real number a | ||
(note that a >= 0 here) |
These rules can be used when working with square roots algebraically. Example.
It is important to notice that, unlike the product property for square roots
Square roots in denominators
When you are adding or terms which contain square roots in a denominator, you may find it helpful to write the expression as a single fraction. This involves using techniques from algebra, such as finding a common denominator, which is shown in the following example. (Here, is the common denominator.)
Similar techniques can be used when the expression under the root sign has more than one term. Example.
Rationalizing
"Rationalizing" a denominator involves eliminating (algebraically) a square root from the denominator of an expression. When the denominator has two terms, we rationalize by multiplying numerator and denominator by the conjugate of the denominator.
In this example, can you identify the conjugate?
Numerators can be rationalized as well.
Other Roots or Radicals
The (principal) n^{th} root of a number a, denoted , is defined as follows: (n is a positive integer here)
where a >= 0 and b >= 0 if n is even, and a and b are real numbers if n is odd.
For example, since 2^{5}=32.
'n' (the root) is sometimes called the index of the radical or root. The expression under the root sign is called the radicand .
Some points about n^{th} roots
- If n is even, then the radical makes sense only if the radicand is positive.
- If n is even, then is defined to be a non-negative number. For example, even though (-3)^{4} as well. So expressions like or produce non-negative quantities.
- If n is odd, there are no restrictions on the radicand; it may be any real number. Examples . In general, the odd root of a negative number is negative, and the odd root of a positive number is positive.
- Roots of powers When taking the nth root of a power, such as ( = 4), you might notice that it is the same as . In fact, = is true for any positive integers m and n, provided they are not both even (and provided the radical exists).
Other roots have properties similar to those for square roots. The properties are listed in this table.