Roots are the inverse of exponents. An *n*th root “undoes” raising a
number to the *n*th power, and vice-versa. (The correct terminology for
these types of relationships is *inverse functions*, but powers and roots
can only be strictly classified as inverse functions if we take care of some
ambiguities associated with plus or minus signs, so we will not worry about
this yet). The common example is the *square root*, which “undoes” the act
of squaring. For example, take 3 and square it to get 9. Now take the square
root of 9 and get 3 again. It is also possible to have roots related to powers
other than the square. The cube root, for example, is the inverse of raising to
the power of 3. The cube root of 8 is 2 because 2^{3} = 8. In general,
the *n*th root of a number is written:

_{} if and only if _{}

_{} because 4^{3} = 64

We leave the index off the square root symbol only because it is the most common one. It is understood that if no index is shown, then the index is 2.

_{} if and only if _{}

_{} because 4^{2} = 16

The square root is the inverse function of squaring (strictly speaking only for positive numbers, because sign information can be lost)

- Every positive number has two square roots, one positive and one negative

**Example:** 2 is a square root of 4 because 2 ´ 2 = 4,
but –2 is also a square root of 4 because (–2) ´ (–2) = 4

To avoid confusion between the two we **define**
the symbol _{}(this symbol is called a *radical*) to mean the **principal**
or **positive** square root.

The convention is:

For any positive number *x*,

_{} is the positive root,
and

_{} is the negative root.

If you mean the negative root, use a minus sign in front of the radical.

**Example:**

_{}

_{} for all non-negative
numbers *x*

_{} for all non-negative
numbers *x*

However, if *x* happens to be
negative, then squaring it will produce a positive number, which will have a
positive square root, so

_{} for all real numbers
*x*

·
You don’t need the absolute value sign if you already
know that *x* is positive. For example, _{}, and saying anything about the absolute value of 2 would be
superfluous. You only need the absolute value signs when you are taking the
square root of a square of a *variable*, which may be positive or
negative.

· The square root of a negative number is undefined, because anything times itself will give a positive (or zero) result.

_{} (your calculator will
probably say ERROR)

· Note: Zero has only one square root (itself). Zero is considered neither positive nor negative.

**WARNING**: Do not
attempt to do something like the distributive law with radicals:

_{} (**WRONG)** or _{} (**WRONG)**. This
is a violation of the order of operations. The radical operates on the *result*
of everything inside of it, not individual terms. Try it with numbers to see:

_{} (**CORRECT**)

But if we (incorrectly) do the square roots first, we get

_{} (**WRONG**)

However, radicals __do__ distribute over products:

_{}

and

_{}

provided that both *a* and *b* are non-negative (otherwise you
would have the square root of a negative number).

Some numbers are perfect squares, that is, their square roots are integers:

0, 1, 4, 9, 16, 25, 36, etc.

It turns out that all other whole numbers have irrational square roots:

_{}, _{}, _{}, _{} etc. are all
irrational numbers.

· The square root of an integer is either perfect or irrational