The square root of a number is the number that, when squared (multiplied
by itself), is equal to the given number. For example, the square root of 16,
denoted
16^{1/2}
or
, is 4, because
4^{2} = 4×4 = 16
. The square
root of 121, denoted
, is 11, because
11^{2} = 121
.
= 5/3
, because
(5/3)^{2} = 25/9
.
= 9
, because
9^{2} = 81
. To take the square root of a
fraction, take the square root of the numerator and the square root of the
denominator. The square root of a number is always positive.

All perfect squares have square roots that are whole numbers. All fractions that have a perfect square in both numerator and denominator have square roots that are rational numbers. For example, = 9/7 . All other positive numbers have squares that are non-terminating, non- repeating decimals, or irrational numbers. For example, = 1.41421356... and = 2.19503572....

Since a positive number multiplied by itself (a positive number) is always positive, and a negative number multiplied by itself (a negative number) is always positive, a number squared is always positive. Therefore, we cannot take the square root of a negative number.

Taking a square root is almost the inverse operation of taking a square. Squaring a positive number and then taking the square root of the result does not change the number: = = 6 . However, squaring a negative number and then taking the square root of the result is equivalent to taking the opposite of the negative number: = = 7 . Thus, we conclude that squaring any number and then taking the square root of the result is equivalent to taking the absolute value of the given number. For example, = | 6| = 6 , and = | - 7| = 7 .

Taking the square root first and then squaring the result yields a slightly
different case. When we take the square root of a positive number and then
square the result, the number does not change:
()^{2} = 11^{2} = 121
. However, we cannot take the square root of a negative
number and then square the result, for the simple reason that it is impossible
to take the square root of a negative number.

A cube root is a number that, when cubed, is equal to the given number.
It is denoted with an exponent of "1/3". For example, the cube root of 27 is
27^{1/3} = 3
. The cube root of 125/343 is
(125/343)^{1/3} = (125^{1/3})/(343^{1/3}) = 25/7
.

Roots can also extend to a higher order than cube roots. The 4th root of a number is a number that, when taken to the fourth power, is equal to the given number. The 5th root of a number is a number that, when taken to the fifth power, is equal to the given number, and so on. The 4th root is denoted by an exponent of "1/4", the 5th root is denoted by an exponent of "1/5"; every root is denoted by an exponent with 1 in the numerator and the order of root in the denominator.

An odd root of a negative number is a negative number. We cannot take an even
root of a negative number. For example,
(- 27)^{1/3} = - 3
, but
(- 81)^{1/4}
does not exist.

We have just learned that a fractional exponent with "1" in the numerator is a root of some sort. But what would an exponent of "2/3" mean? Or an exponent of " -5/2 "?

In a fractional exponent, the numerator is the power to which the number
should be taken and the denominator is the root that should be taken. For
example,
64^{2/3}
means "square 64 and take the cube root of the
result" or "take the cube root of 64 and square the result. This works out to
16.

A negative fractional exponent works just like a negative exponent. First,
we switch the numerator and the denominator of the base number, and then
we apply the positive exponent. For example,
(9/25)^{-5/2} = (25/9)^{5/2} = (25^{5/2})/(9^{5/2}) =
"the square root
of 25 to the fifth power over the square root of 9 to the fifth power"
= 3, 125/243
.
27^{-1/3} = (1/27)^{1/3} = (1^{1/3})/(27^{1/3}) = 1/3
.

Again, we cannot take a negative number to a fractional power if the denominator of the exponent is even.