small figure is placed above, a little to the right of the number whose power is to be found. The small figure is called the index, or exponent. Thus, 4-4x4-16; here the exponent is 2, and 42 denotes the second power of 4. 31: == 32-3x3= In the same way we have 3 the first power of 3. 333x3x3 27 the third power of 3. = 34=3x3x3x3= 81 the fourth power of 3. 35=3x3x3x3x3=243 the fifth power of 3. &c., The second power of a number is called the square of that number, because it may be represented by means of a geometrical square. Thus, in the adjacent figure if the side of this square is 12 linear units, as 12 inches long, its entire surface will be denoted by 12 x 12 144 square units, which in this case will be 144 square inches. For a similar reason, the third power of a number is called the cube of that number, since it can be represented by the geometrical cube, as in the adjacent figure, where the side of the cube is supposed to be 3 linear feet, consequently each face will be 3x3=9 square 3 feet. 3 feet. feet, and its volume will be 3 x3x3=27 cubic feet. ing To raise a number to any power, we have the follow RULE. Multiply the number continually by itself, as many times less one as there are units in the exponent; the last product will be the power sought. What is Involution? How do we denote that a number is to be raised to a power? What is this small figure placed above, a little to the right, called? Repeat the Rule. 12. What is the cube power of 34? Ans. 1oo=377. 13. What is the fifth power of 23? Ans. 10101157,283. 14. What is the third power of 0.5? 16. What is the square of ? 17. What is the cube of 1? 18. What is the cube of 21? Ans. 0.125. EVOLUTION. 131. EVOLUTION is the reverse of involution; that is, it explains the method of resolving a number into equal factors. When a number can be resolved into equal factors, one of these factors is called a root of the number. If the number is resolved into two equal factors, one of these factors is called the square root. Thus, 36 6×6, and 6 is the square root of 36. In the same way 7 is the square root of 49, since 49=7×7. To denote that the square root of a number is to be found, we use the symbol V. Thus, 81 denotes that the square root of 81 is to be found; that is, √/81=9; so √100=10; √/25=5. When a number is resolved into three equal factors, one of these factors is called the cube root of the number. Thus, 644×4 × 4, and 4 is the cube root of 64; also 5 is the cube root of 125, since 125-5×5×5. To indicate that the cube root of a number is to be found, we use the symbol ; thus, 27 denotes that the cube root of 27 is to be found; that is, 27=3; so 64 =4; 8=2; 216-6. We shall hereafter use the dot (.) to denote multiplica tion. Thus 3.4 indicates that 3 is to be multiplied by 4. Also 3 × 4.8 denotes that the product of 3 and 4 is to be multiplied by 8. When the dot is used to denote multiplication, it is placed near the bottom of the line, but when used to denote a decimal, it is placed near the middle of the line. What is Evolution? When a number can be resolved into a number of equal factors, what is such a factor called? If the number is resolved into two equal factors, what is the root called? When resolved into three equal factors, what is the root called? What character is used to denote the square root? What to denote the cube root? What is the square root of 81? What is the square root of 100? What is the cube root of 27? What is the cube root of 8? What additional sign of mul tiplication is used? Before explaining the method of extracting the square roots of numbers, we shall involve some numbers by considering them as decomposed into units, tens, hundreds, &c. What is the square of 25? Of 35? By a similar method, we find 46 (40+6)=402+2x40.6+6=1600+480+36. 54 (50+4)=502+2x50.4+42=2500+400+16. 932 (90+3)=902+2×90.3+32=8100+540+ 9. 482 (40+8)=402+2×40.8+8=1600+640+64. From the above, we draw the following property: The square of the sum of two numbers is equal to the square of the first number, plus twice the product of the first number into the second, plus the square of the second. If we wish the square of the sum of three numbers, as 6+8+9; we may unite the first and second by means of a parenthesis, thus, for 6+8+9, we may make use of (6+8)+9; and now regarding 6+8 as one number, the preceding rule for the sum of two numbers will apply to (6+8)+9, that is, the square of 6+8+9 is equal to the square of (6+8,) plus twice the product of (6+8) into 9, plus the square of 9. But the square of 6+8 has already been shown to be, the square of 6, plus twice the product of 6 into 8, plus the square of 8. Hence, the square of 6+8+9 is equal to the square of 6, plus twice the product of 6 into 8, plus the square of 8, plus twice the product of the sum of 6 and 8 into 9, plus the square of 9. Or in general terms, The square of the sum of three numbers is equal to the square of the first number, plus twice the product of the first number into the second, plus the square of the second; plus twice the product of the sum of the first two into the third, plus the square of the third. Continuing in this way, we could show that, the square of the sum of any number of numbers is the square of the first number, plus twice the product of the first number into the second, plus the square of the second; plus twice the product of the sum of the first two into the third, plus the square of the third; plus twice the product of the sum of the first three into the fourth; plus the square of the fourth; plus twice the product of the sum of the first four into the fifth, plus the square of the fifth; and so on. We will now apply this general rule to a few examples. 1. (2+3)2=22+2×2.3+3o. 2. (5+7)2=52+2×5.7+7o. 3. (3+4+5)2=3o+2×3.4+4°+2×(3+4).5+5o. 4. (5+6+7)2=52+2×5.6+62+2×(5+6).7+7o. 5. (7+8+9)2=72+2×7.8+82+2×(7+8).9+92. 6. (35)2=(30+5)2=302+2×30.5+5o. 7. (47)3=(40+7)=40+2X40,7+7%. 8. (365)=(300+60+5)=3002+2x 300.60+603+ 2x(300+60).5+5% |