EXAMPLE. The letters having the same values as before, what does EXAMPLES FOR PRACTICE. 29. Find the numerical values of x in the following formulas, when a = 9, b = 8, c = 2, d = 10, and e = = 3: d+ce (a) x = bd-40 f(a + e) се 126D-10 Ans. . . Ans. x = 1}. Ans. x = 166.5. Ans. x 21. INVOLUTION. 30. If a product consists of equal factors, it is called a power of one of those equal factors, and one of the equal factors is called a root of the product. The power and the root are named according to the number of equal factors in the product. Thus, 3 x 3, or 9, is the second power, or square, of 3; 3 x 3 x 3, or 27, is the third power, or cube, of 3; 3 × 3 × 3 × 3, or 81, is the fourth power of 3. Also, 3 is the second root, or square root, of 9; 3 is the third root, or cube root, of 27; 3 is the fourth root of 81. 31. For the sake of brevity, 3 X 3 is written 3', and read three square, or three exponent two; 3 x 3 x 3 is written 3', and read three cube, 3 × 3 × 3 × 3 is written 3*, and read three fourth, and so on. or three exponent four; A number written above and to the right of another number, to show how often the latter number is used as a factor, is called an exponent. Thus, in 3", the number" is the exponent, and shows that 3 is to be used as a factor twelve times; so that 3" is a contraction for 3X 3X 3X 3X 3X 3X 3X3x3x3 x 3 x 3. In an expression like 3', the exponent shows how often 3 is used as a factor. Hence, if the exponent of a number is unity, the number is used once as a factor; thus, 3' 4' 4, 5' 5. = = 3, 32. If the side of a square contains 5 inches, the area of the square contains 5 × 5, or 5', square inches. If the edge of a cube contains 5 inches, the volume of the cube contains 5x5x5, or 5', cubic inches. It is for this reason that 5' and 5' are called the square and cube of 5, respectively. 33. To find any power of a number: EXAMPLE 1.-What is the third power, or cube, of 35? SOLUTION. cube 1.7 28 Ans. EXAMPLE 4.-What is the third power, or cube, of? 34. Rule.-I. To raise a whole number or a decimal to any power, use it as a factor as many times as there are units in the exponent. II. To raise a fraction to any power, raise both the numer ator and denominator to the power indicated by the exponent. ARITHMETIC. (PART 6.) MENSURATION. 1. Mensuration treats of the measurement of lines, angles, surfaces, and solids. LINES AND ANGLES. 2. A straight line is one that does not change its direction throughout its whole length-it is the shortest distance between two points.. To distinguish one straight line from B FIG. 1. another, two of its points are designated by letters. The line shown in Fig. 1 would be called the line A B. 3. A curved line changes its direction at every point. (Fig. 2.) B A FIG. 2. 4. Parallel lines are equally distant from each other at all points. (Fig. 3.) 5. A line is perpendicular to another when it meets that line so as not to incline towards it on either side. (Fig. 4.) 6. A horizontal line is a line parallel to the horizon or water level. (Fig. 5.) 7. A vertical line is a line perpendicular to a horizontal line; consequently, it has the direction of a plumb-line. (Fig. 5.) FIG. 3. FIG. 4. Vertical. Horizontal. For notice of copyright, see page immediately following the title page. |