4. It is required to find the square root of 4x-4x2+ 12x3+x2-6x+9. Ans. 2x3x+3. 5. Required the square root of x+4x5+10x1+20x3 +25x2+24x+16. Ans. x3+2x2+3x+4. 6. It is required to extract the square root of a2 +b. Ans. a+ b ba 63 + 564 2a 8a3 16a5 12807, &c. 7. It is required to extract the square root of 2, or of Ans. 1++ √ √1⁄2 + √, &c. 1+1. CASE III. - 32 To find any root of a compound quantity. RULE. Find the root of the first term, which place in the quotient; and having subtracted its corresponding power from that term, bring down the second term for a dividend. Divide this by twice the part of the root above determined, for the square root; by three times the square of it, for the cube root, and so on; and the quotient will be the next term of the root. Involve the whole of the root, thus found, to its proper power, which subtract from the given quantity, and divide the first term of the remainder by the same divisor as before; and proceed in this manner till the whole is finished*. * As this rule, in high powers, is often found to be very laborious, it may be proper to observe, that the roots of various compound quantities may sometimes be easily discovered, as follows: Extract the roots of all the simple terms. and connect them together by the signs or, as may be judged most suitable for the purpose; then involve the compound root, thus found, to its proper power, and if it be the same with the given quantity, it is the root required. But if it be found to differ only in some of the signs, change them from to, or from -to +, till its power agrees with the given one throughout. Thus, in the third example next following, the root is 2a-3x, which is the difference of the roots of the first, and last terms; and in the fourth example, the root is a+b+c, which is the sum of the roots of the first, fourth, and sixth terms. The same may also be observed of the sixth example, where the root is found from the first and last terms. EXAMPLES. 1. Required the square root of a4 -2a3x+3a2x2-2ax 3 64. 2. Required the cube root of x+6x5 −40x3 +96x— x-6x540x3+96x-64(x+2x-4 x6 3x+(6x5 x+6x5+12x4+8x3 3x4) — 12x4 x+6x5-40x3+96x-64 * 3. Required the square root of 4a2-12ax+9x2. Ans. 2a-3x. Ans. a+b+c. 4. Required the square root of a2+2ab+2ac+b2+ 2bc+c2. 5. Required the cube root of x-6x515x1-20x3+ 15x2-6x+1. Ans. x2-2x+1. 6. Required the 4th root of 16a — 96a3x+216a2x2 216ax381x4. Ans. 2a-3x. 7. Required the 5th root of 32x5 - 80x1+80x3 —40x2 +10x-1. Ans. 2x-1. OF IRRATIONAL QUANTITIES, OR SURDS. IRRATIONAL Quantities, or Surds, are those of which the values cannot be accurately expressed in numbers; and are usually expressed by means of the radical sign✔, or by fractional indices; in which latter case, the numerator shows the power the quantity is to be raised to, and the denominator its root. Thus, 1/2, or 2, denotes the square root of 2; 3⁄4/a3, or a3. the cube root of the square of a, &c.* CASE I. To reduce a rational quantity to the form of a surd. RULE. Raise the quantity to a power corresponding with that denoted by the index of the surd; and over this new quantity place the radical sign, or proper index, and it will be of the form required. EXAMPLES. 1. Let 3 be reduced to the form of the square root, Here 3X3=32=9; whence √9. Ans. * A quantity of the kind here mentioned, as for instance ✔2, is called an irrational number, or a surd, because no number, either whole or fractional, can be found, which when multiplied by itself, will produce 2. But its ap proximate value may be determined to any degree of exactness, by the common rule for extracting the square root, being 1 and certain non periodic de cimals, which never terminate. 2. Reduce 2x2 to the form of the cube root. 3 Here (2x2)3=8x; whence 3/8x®, or (8x€)3. 3. Let 5 be reduced to the form of the square root. Ans. (25). 4. Let -3x be reduced to the form of the cube root. Ans. (27x3). 5. Let-2a be reduced to the form of the fourth root. Ans. (16a). 6. Let a2 be reduced to the form of the fifth root, and va ✔a+vb, and to the form of the square root. 2a ba a Ansa1o, √(a+2 √ ab+b), √ (¦ α), and √52. Note. Any rational quantity may be reduced by the above rule, to the form of the surd to which it is joined, and their product be then placed under the same index, or radical sign. EXAMPLES. Thus 2/2=√4×√2=√4×2=√8 4a/}{/4a=2/X4a=3/g Ans. (150). 2. Let 5a be reduced to a simple radical form. Ans. (*). To reduce quantities of different indices, to others that shall have a given index. RULE. Divide the indices of the proposed quantities by the given index, and the quotients will be the new indices for those quantities. Then, over the said quantities, with their new indices, place the given index, and they will be the equivalent quantities required. EXAMPLES. 1. Reduce 3 and 23 to quantities that shall have the index. 3 6 3 1 3 6 1 Whence (33) and (22), or 27 and 4*, are the quan tities required. 2. Reduce 5 and 63 to quantities that shall have the Ans. 125 and 36%. common index 1 6 3. Reduce 21 and 4 to quantities that shall have the common index 27 1 8 Ans. 16 and 16. 4. Reduce a2 and a to quantities that shall have the common index Ans. (a) and (a2). 5. Reduce a and b to quantities that shall have the common index Ans. (a3)3 and (b1)*. 1 6 Note. Surds may also be brought to a common index, by reducing the indices of the quantities to a common denominator, and then involving each of them to the power denoted by its numerator. |