Practice Problem: Evaluate the expression in each case. We now also have a sufficient foundation to allow us to relate roots to exponents. We will deal primarily with square roots you should now, however, have some familiarity with other roots. Furthermore, very few numbers have k th roots that are integers, meaning that most k th roots are irrational numbers, which cannot be written exactly in decimal or fractional form. Some further examples are shown below.Ĭalculating roots beyond the square root mentally or by hand gets increasingly difficult thus, in such cases, a calculator is generally needed. In other words, if some number n is equal to a number r raised to the power of k, then the k th root of n is r. We can also define a general k th root of a number n that is, These expressions relate the number n and its cube root r. To represent the third root, we add a small number 3 next to the radical as shown in the relations below. We can also calculate, for instance, the third root (also called the cube root) of a number n. To reiterate, the square root of n is a number r, where the following relations apply. In some instances, however, we might be interested in calculating other roots of numbers. For example, we might prefer to write instead of 1.414, which is only an approximation to three decimal places.Ībove, we discussed square roots exclusively, and we noted that the radical symbol by itself indicates the square root. Thus, in the interest of exact math, leaving the numbers in radical form (instead of writing an approximate decimal) is sometimes preferable. Thus, calculators (including computers) and long methods of calculating the square root can only give approximate results the actual numbers have an infinite number of decimal places and cannot be written as a fraction with integers in the numerator and denominator. Furthermore, the square roots of many numbers (including integers that are not perfect squares) are irrational. Long ("by hand") methods for calculating a square root do exist, however, although they can be tedious. Generally, evaluating square roots requires a calculator.
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