A root is a number that when multiplied by itself a specified number of times will produce a given number.

The two most common roots are the square root and the cube root. For more examples of roots, see the chart in Figure 1-10, Functions of Numbers.

Square Roots

The square root of 25, written as √25, equals 5. That is, when the number 5 is squared (multiplied by itself ), it produces the number 25. The symbol √ is called a radical sign. Finding the square root of a number is the most common application of roots. The collection of numbers whose square roots are whole numbers are called perfect squares. The first ten perfect squares are: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. The square root of each of these numbers is 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10, respectively.

For example, √36 = 6 and √81 = 9

To find the square root of a number that is not a perfect square, use either a calculator or the estimation method. A longhand method does exist for finding square roots, but with the advent of calculators and because of its lengthy explanation, it is no longer included in this handbook. The estimation method uses the knowledge of perfect squares to approximate the square root of a number.

Example: Find the square root of 31. Since 31 falls between the two perfect roots 25 and 36, we know that √31 must be between √25 and √36. Therefore,√31 must be greater than 5 and less than 6 because √25 = 5 and √36 = 6. If you estimate the square root of 31 at 5.5, you are close to the correct answer. The square root of 31 is actually 5.568.

Cube Roots

The cube root of 125, written as 3√125, equals 5. That is, when the number 5 is cubed (5 multiplied by itself then multiplying the product (25) by 5 again), it produces the number 125. It is common to confuse the “cube” of a number with the “cube root” of a number. For clarification, the cube of 27 = 273 = 27 × 27 × 27 = 19,683. However, the cube root of 27 = 3√27 = 3.

Fractional Powers

Another way to write a root is to use a fraction as the power (or exponent) instead of the radical sign. The square root of a number is written with a 1⁄2 as the exponent instead of a radical sign. The cube root of a number is written with an exponent of 1⁄3 and the fourth root with an exponent of 1⁄4 and so on.


Figure 1-10a. Functions of numbers.

Figure 1-10a. Functions of numbers.

Figure 1-10b. Functions of numbers.

Figure 1-10b. Functions of numbers.