Powers, Exponents, and Roots
Square Roots
Square Roots
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
161/2
or
, is 4, because
42 = 4×4 = 16
. The square
root of 121, denoted
, is 11, because
112 = 121
.
= 5/3
, because
(5/3)2 = 25/9
.
= 9
, because
92 = 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....
Square Roots of Negative Numbers
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 = 112 = 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.
Cube Roots and Higher Order Roots
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 271/3 = 3 . The cube root of 125/343 is (125/343)1/3 = (1251/3)/(3431/3) = 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.
Fractional Exponents
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, 642/3 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 = (255/2)/(95/2) = "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 = (11/3)/(271/3) = 1/3 .
Again, we cannot take a negative number to a fractional power if the denominator of the exponent is even.
