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 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....

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 27^1/3 = 3. The cube root of 125/343 is (125/343)^1/3 = (125^1/3)/(343^1/3) = 25/7.