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.

means that x2=a, for x >= 0.

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

there is no similar rule for a sum under a square root. That is, in general

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" 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) nth root of a number a, denoted , is defined as follows: (n is a positive integer here)

means a = bn

where a >= 0 and b >= 0 if n is even, and a and b are real numbers if n is odd.

For example, since 25=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 nth roots

Other roots have properties similar to those for square roots. The properties are listed in this table.

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