POWERS, ROOTS, EXPONENTS, and radicals

Figure 1-2.-2-percent grade. In connection with the study of decimal fractions, businessmen as early as the fifteenth century made use of certain decimal fractions so much that they gave them the special designation PERCENT. The word percent is derived from Latin. It was originally per centum, which means “by the hundredths.” In banking, interest rates are always expressed in percent; statisticians use percent; in fact, people in almost all walks of life use percent to indicate increases or decreases in production, population, cost of living, and so on. The Engineering Aid uses percent to express change in grade (slope), as shown in figure 1-2. Percent is also used in earthwork computations, progress reports, and other graphical representa- tions. Study chapter 6 of NAVEDTRA 1-0069-D1 for a clear understanding of percentage. POWERS, ROOTS, EXPONENTS, AND RADICALS Any number is a higher power of a given root. To raise a number to a power means to multiply, using the number as a factor as many times as the power indicates. A particular power is indicated by a small numeral called the EXPONENT; for example, the small 2 on 3²is an exponent indicating the power. Many formulas require the power or roots of a number. When an exponent occurs, it must always be written unless its value is 1. A particular ROOT is indicated by the radical sign (~), together with a small number called the INDEX of the root. The number under the radical sign is called the RADICAND. When the radical sign is used alone, it is generally understood to mean a square root, and ~, ~, and ~, indicate cube, fifth, and seventh roots, respec- tively. The square root of a number may be either + or – . The square root of 36 may be written thus: G = t6, since 36 could have been the product of ( + 6)( + 6) or ( – 6)( – 6). However, in practice, it is more convenient to disregard the double sign ( ± ). This example is what we call the root of a perfect square. Sometimes it is easier to extract part of a root only after separation of the factors of the number, such as: ~ = ~ = 3~. As you can see, we were able to extract only the square root of 9, and 3 remains in the radical because it is an irrational factor. This simplification of the radical makes the solution easier because you will be dealing with perfect squares and smaller numbers. Examples: Radicals are multiplied or divided directly. Examples: Like fractions, radicals can be added or sub- tracted only if they are similar. Examples: When you encounter a fraction under the radical, you have to RATIONALIZE the denominator before performing the indicated operation. If you multiply the numerator and denominator by the same number, you can 1-4