801. PRINCIPLE.-The sum of the exponents of two powers of the same number is equal to the exponent of the product of those powers. Thus, 22 × 23-25; for 22=2 × 2, and 23=2 × 2 × 2; hence 22 × 23= 2 × 2 × 2 × 2 × 2=25. 802. To find any power of a number. 1. Find the third power of 35. 35= OPERATION. ANALYSIS. Since using 351; 35 x 35 = 352 = 1225 any number three times as a factor produces the third power of that num Of 42. Of 56. Of 75. Of 42. Of 54. 2. Find the square of 37. 3. Find the cube of 15. Of 18. 4. What is the value of 632? of 483? of 324 ? of 125? RULE. Find the product of as many factors, each equal to the given number, as there are units in the exponent of the required power. 5. What is the third power of ? RULE.-A fraction may be raised to any power by involving each of its terms separately to the required power. 6. What is the square of? 7. Raise to the 4th power. The cube of ? Find the value of each of the following expressions: FORMATION OF SQUARES AND CUBES BY THE ANALYT ICAL METHOD. 803. To find the square of a number in terms of its tens and units. 1. Find the square of 27 in terms of its tens and units. PRINCIPLE.-The square of a number consisting of tens and units is equal to the sum of the squares of the tens and units increased by twice their product. 3. Find the square of 42 in terms of its tens and units. In like manner find the square 804. To find the cube of a number in terms of its tens and units. 1. Find the cube of 25 in terms of its tens and units. 258203+ (3 × 202 × 5) + (3 × 20 × 52)+58 ANALYSIS.-The square of 25 is 202 + (2 × 20 × 5) +5o. (803, PRIN.) Multiplying this by 20+5 gives the cube of 25. 2. Find the cube of 34 in terms of its tens and units. PRINCIPLE.-The cube of a number consisting of tens and units is equal to the cube of the tens, plus three times the product of the square of the tens by the units, plus three times the product of the tens by the square of the units, plus the cube of the units. GEOMETRICAL ILLUSTRATION. The volume of the cube marked A, Fig. 1, is 203; the volume of each of the rectangular solids marked B is 20 x 20 x 5, or 202 × 5; the volume of each of the rectangular solids marked C, in Fig. 2, is 20 × 5 × 5, or 20 x 52; and the volume of the small cube marked D is 53. It is evident, that if all these solids are put together as represented in Fig. 3, a cube will be formed, each edge of which is 25. 3. Find the cube of 46 ? OPERATION. 403-64000 402 × 6 × 3=28800 40 x 62 x 3 = 4320 EVOLUTION 805. 1. What are the two equal factors of 25? 36? 2. What are the three equal factors of 27? 64? 125? 3. What are the four equal factors of 16? 81? 256? 4. Of what is 81 the 2d power? The 4th power? DEFINITIONS. 806. The Square Root of a number is one of the two equal factors of that number; the Cube Root is one of the three equal factors of that number, etc. Thus, 3 is the square root of 9, 2 is the cube root of 8, etc. 807. Evolution is the process of finding the root of any power of a number. 808. The Radical Sign is √. When prefixed to a number, it indicates that some root of it is to be found. 809. The Index of the root is a small figure placed above the radical sign to denote what root is to be found. When no index is written, the index 2 is understood. Thus, 100 denotes the square root of 100; 125 denotes the cube root of 125; 256 denotes the fourth root of 256; and so on. Evolution, or both involution and evolution, may be indicated in the same expression by a fractional exponent, the numerator denoting the required power of the given number, and the denominator the root of that power of the number. Thus, 9 is equivalent to /9; 643, to /64; and 83, to the cube root of the second power of 8, equivalent to /83, etc. |