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 mised 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 37? Ans. 1o=377. 13. What is the fifth power of 24? Ans. 18101-157-284. 14. What is the third power of 0.5? 16. What is the square of ? 17. What is the cube of 11? 18. What is the cube of 21? Ans. 0.125. Ans. 0.00390625. Ans. Ans. 33. Ans. 10,8. 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; When a number is resolved into three equal factors, one of these factors is called the cube root of the number. Thus, 64=4×4 × 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; so64 =4; 8=2; 216-6. We shall hereafter use the dot (.) to denote multiplication. Thus 3.4 indicates that 3 is to be multiplied by 4. Also 3x4.8 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 fao tors, 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? = 352-900+300+ 25 46° (40+6)=40+2 x 40.6+61600+480+36. 542 (50+4)=502+2x50.4+4=2500+400+16. 93 (90+3)=902+2x90.3+32=8100+540+ 9. 482 (40+8)=40+2x40.8+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 cf (6+8) into 9 plus the square of 9. But the square of +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 、o 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 ex amples. 1. (2+3)2=22+2×2.3+32. 2. (5+7)=52+2×5.7+7%. 3. (3+4+5)2=32+2×3.4+42+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+9o. 6. (35)2=(30+5)=302+2×30.5+5o. 7. (47)2=(40+7)=402+2×40.7+7o. 8. (365)=(300+60+5)2=3002+2× 300.60+602+ 2x(300+60).5+5. 9. (487)2=(400+80+7)=4002+2× 400.80 +802+ 2x (400+80).7+72. The above method of squaring a number consisting of the sum of two or more numbers, is elegantly illustrated geometrically as follows: The square ABCD may be enarged to the square AEKF, by the addition of the two equal rectangles BG and DH, whose lengths are each equal to the side AB of the original square, and whose widths are equal to BE, the quantity by which the side of the square has been augmented, also a little square, CGKH, whose side is the same as the width of one of the equal rectangles. M Р F (400+80) 7 72R H K 400-80 802 D C A 4002 (400-+-80)-7 400-80 Again, the square AEKF, having its side increased by EL, becomes augmented by the two rectangles EN, FP, and the little square KR. Thus we might continue to augment the square last found by the addition of two equal rectangles, and a little square; the length of each rectangle being equal to the side of the square which is to be augmented, and the width equal to the quantity by which the side of the square is increased; and the side of the little square being the same as the width of one of the rectangles. The diagram is adapted to the case of squaring 400+80+7=487. 132. Let us now, by reversing the above process, deduce a rule for extracting the square root. Let it be required to extract the square root of 527076. For the sake of simplicity, we will suppose we are required to find the number of feet in the side of a square whose area shall contain 527076 square feet. The smallest number, consisting of two figures, which is 10, becomes, when squared, 100; having more than two figures. Again, the largest number of two figures, 99, becomes, when squared, 9801, having four figures. Hence, when a number consists of more than two figures, and of not more than four, its square root will consist of two figures. By a similar method it may be shown, that when a number consists of more than four, and of not more than six |