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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
.

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