Powers, Exponents, and 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 16^1/2 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....

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.

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.