5. Expand (2a + 7)3. Ans. 8a84 a2 + 294 a + 343. 6. Expand (2 ac — x)1. Ans. 16ac32 a3 c3 x+24 a2c2x2-8 ac x2+x2. 7. Expand (a2 - 2 y)3. x 131. The Binomial Theorem can be applied to the expansion of a polynomial. Thus, in ab. y can C, a+b can be treated as a single term, and the quantity can be written (a+b) - C. In like manner, a+b+x be written (a + b) + (x − y). In such cases it is easier to substitute a single letter for the enclosed terms, and after the expansion to substitute the proper values. Substituting for x, its value, a+b; (a + b −c)3 = a3 + 3 a2b+3a b2 + b3 — 3 a2 c-6 a b c -352 c + 3 ac2 + 3bc2 — c3 2. Expand (2a-b-c-d)2. NOTE. For 2 a-b-c-d write (2a-b)-(c+d). Ans. 4a-4ab+b2-4ac-4ad+2bc+2bd+c2+2cd+d3. 3. Expand (3x-ya+b)2. 4. Expand (x — a + b)3. EVOLUTION. 132. EVOLUTION is the process of extracting a root of a quantity. It is the reverse of Involution. 133. A ROOT of any quantity is a quantity which taken. as a factor a given number of times will produce the given quantity. The number of times the root is to be taken as a factor depends upon the name of the root. Thus, the second or square root of a quantity is a quantity which taken twice as a factor will produce the given quantity; the third or cube root is a quantity which taken three times as a factor will produce the given quantity; and so on. A RooT is indicated by the radical sign, or by a fractional exponent. Thus, ✔, or indicates the square root of x. 134. A root and a power may be indicated at the same time. Thus, x, or x, indicates the cube root of the fourth power of x, or the fourth power of the cube root of x; for a power of a root of a quantity is equal to the same root of the same power of the quantity. ✅82 or 83 is the square of the cube root of 8, or the cube root of the square of 8, i. e. 4. 135. A perfect power is a quantity whose root can be found. A perfect square is one whose square root can be found; a perfect cube is one whose cube root can be found; and so on. 136. Since Evolution is the reverse of Involution, the rules for Evolution are derived at once from those of Involution. And therefore, as according to Art. 125 an odd power of any quantity has the same sign as the quantity itself, and an even power is always positive, we have for the signs in evolution the following RULE. An odd root of a quantity has the same sign as the quantity itself. An even root of a positive quantity is either positive or negative. An even root of a negative quantity is impossible, or imaginary. SQUARE ROOT OF NUMBERS. 137. THE SQUARE ROOT of a number is a number which, taken twice as a factor, will produce the given number. 138. The square of a number has twice as many figures as the root, or one less than twice as many. Thus, The square of any number less than 10 must be less than 100; but any number less than 10 is expressed by one figure, and any number less than 100 by less than three figures; i. e. the square of a number consisting of one figure is a number of either one or two figures. The square of any number between 10 and 100 must be between 100 and 10000; i. e. must contain more than two figures and less than five. And the square of any number between 100 and 1000 must contain more than four figures and less than seven. Hence, to ascertain the number of figures in the square root of a given number, Beginning at units, point off the number into periods of two figures each; there will be as many figures in the root as there are periods, and for the incomplete period at the left, if any, one more. 139. To extract the square root of a number. 1. Find the square root of 5329. From the preceding explanation, it is evident that the square root of 5329 is a number of two figures, and that the tens figure of the root is the square root of the greatest perfect square in 53; i. e. √49, or 7. Now, if we represent the tens of the root by a and the units by b, a+b will represent the root; and the given number will be If therefore 429 is divided by 2 a + b, it will give b the units of the root. But bis unknown, and is small compared with 2a; we can therefore use 2 a = 140 as a trial divisor. 429140, or 42143, a number that cannot be too small but may be too great, because we have divided by 2a instead of 2a + b. Then 63, and 2 a + b = 140 + 3 143, the true divisor; and 429; and therefore 3 is the unit figure of required root. The work will appear as = (2a + b) b= 143 X 3 the root, and 73 is the = Hence, to extract the square root of a number, RULE. Separate the given number into periods of two figures each, by placing a dot over units, hundreds, &c. Find the greatest square in the left-hand period, and place its root at the right. Subtract the square of this root figure from the left-hand period, and to the remainder annex the next period for a dividend. Double the root already found for a TRIAL DIVISOR, and, omitting the right-hand figure of the dividend, divide, and place the quotient as the next figure of the root, and also at the right of the trial divisor for the TRUE DIVISOR. Multiply the true divisor by this new root figure, subtract the product from the dividend, and to the remainder annex the next period, for a new dividend. Double the part of the root already found for a trial divisor, and proceed as before, until all the periods have been employed. |