In a similar manner, polynomials may sometimes be simplified. Thus, √(2a3—4a2b+2ab2)=√/2a(a2—Qab+b2)=(a—b)√2a. A fractional radical of the second degree may be reduced to its simplest form, by the same rule, by first multiplying both terms by any quantity that will render the denominator a perfect square; separating the fraction into two factors, one of which is a perfect square, then extracting the square root of the square factor, and placing it before the other factor placed under the radical sign. 17. Reduce to its simplest form. √}×6=√§×√6={√6. Ans. Reduce the following fractional radicals to their simplest forms. Since a=√/a2, and 2√/3=√4×√/3=√/4×3=√12, it is obvious, that any quantity may be reduced to the form of a radical of the second degree, by squaring it, and placing it under the radical sign. By the same principle, the coëfficient of a radical may be passed under the radical sign. 26. Reduce 5 to the form of a radical of the second degree. Ans. 25. 27 Reduce 2a to the form of a radical of the second degree. Ans. 1/4a2. 28. Express the quantity 37/5, entirely under the radical. Ans. √45 29. Pass the coefficient of the quantity 3c/2c, under the radical. Ans. 18. 30. Pass the coefficient of the quantity 51/3 under the radical. Ans. 175. ADDITION OF RADICALS OF THE SECOND DEGREE. ART. 200.-1. What is the sum of 3√⁄2 and 5√⁄2? It is evident, that 3 times and 5 times any certain quantity must make 8 times that quantity, therefore 3√2+5√/2=8√/2. In the same manner, √2+√/8=√/2+2√/2==3√/2. 2. What is the sum of 21/3 and 51/7? Since dissimilar quantities can not be collected into one sum, we can only add these expressions by placing the sign of addition between them; that is, the sum of 21/3 and 51/7=2√3+5√7. Hence, the RULE, FOR THE ADDITION OF RADICALS OF THE SECOND DEGREE. 1st. Reduce the radicals to their simplest form. 2d. Then, if the radicals are similar, prefix the sum of their coëf ficients to the common radical; but, if they are not similar, connect them by their proper signs. Find the sum of the radicals in each of the following examples. Ans. 2ay/2a. 18. Find the sum of √(2a3—4a2c+2ac2) and √(2a3+4a2c+2ac2). 19. Find the sum of √a+x+√ax2+x3+√(a+x)3. Ans. (1+a+2x)va+x. SUBTRACTION OF RADICALS OF THE SECOND DEGREE. ART. 201.-1. Take 3√2 from 5√2. It is evident that 5 times any quantity minus 3 times the quantity, will be equal to 2 times the quantity, therefore 51/2-3√2-2√2. In the same manner, √/8-√2=2√2−√2=√2. REVIEW.-199. In what does reduction of radicals of the second degree consist? On what principle is it founded? Prove this principle. What is the rule for the reduction of a radical of the second degree to its simplest form? How do you determine if any quantity contains a numerical factor that is a perfect square? How may a fractional radical of the second degree be reduced to its simplest form? 200. What is the rule for the addition of radicals of the second degree? If the radicals are dissimilar, it is obvious that their difference can only be indicated. Thus, if it be required to take 3√a from 5√b, the difference would be expressed by 5√b—3√ā. From these illustrations, we derive the RULE, FOR THE SUBTRACTION OF RADICALS OF THE SECOND DEGREE. 1st. Reduce the radicals to their simplest form; then subtract their coefficients, and prefix the difference to the common radical. 2d. If the radicals are not similar, indicate their difference by the proper sign. 17. From √3m2x+6mnx+3n2x take √/3m2x-6mnx+3n2x. Ans. 2n√/3x. MULTIPLICATION OF RADICALS OF THE SECOND DEGREE. ART. 202.—Since √ab=√α×√b, therefore √aX√b=√ ab. See Art. 199. Also, a√bXc√d=aXcX√bXvd=ac√/bd. From which we have the RULE, FOR THE MULTIPLICATION OF RADICALS OF THE SECOND DEGREE. 1st. Multiply the quantities under the radical sign together, and place the result under the radical. 2d. If the radicals have coëfficients, place their product as a coëfficient before the radical sign. EXAMPLES. 1. Find the product of 6 and 1/8. √6X√8=√48=√16×3=4√3. Ans. 2. Find the product of 2/14 and 3√/2. 2/14/3√/2=67/28-61/4X7=6X27/7=127/7. Ans. 3. Find the product of √/8 and √2. 9. Find the product of 2/15 and 31/35. 10. Find the product of va3bic and abc. 11. Find the product of 12. Find the product of √ and 3. Ans. 4. Ans. Ga. Ans. 9. Ans. 61/6. Ans. 18. Ans. 3/10. . Ans. 30/21. Ans. a2b3c. and √§. Ans. 45. When two polynomials contain radicals of the second degree, they may be multiplied together, in the same manner as in multiplication of polynomials, Art. 72, attending, at the same time, to the directions contained in the preceding rule. 14. Find the product of 2+1/2 and 2-1/2. 15. Find the product of 1+1/2 and 1-1/2.. 16. Find the product of x+2 by x-2. 17. Find the product of va+x by √a+x.. 18. Find the product of √ab+bx by √ ab−bx. A. √a2b2—b2x2. 19. Find the product of √x+2 by vx+3. Ans. √x2+5x+6. Perform the operations indicated in the following examples. 20. (cva+dv/b)X(cva—d√/b). 21. (7+2√6)X(9—5√/6). DIVISION OF RADICALS OF THE SECOND DEGREE. ART. 203. Since Division is the reverse of Multiplication, and since √ã×√b=√ab, therefore √ ab÷√a= √√ab=√. REVIEW.-201. What is the rule for the subtraction of radicals of the second degree? 202. What is the rule for the multiplication of radicals of the second degree? On what principle does it depend? Also, since a√/bXc√/d=ac1/bd, therefore ac√/bd÷ay/bac√/bd αντ a RULE, FOR THE DIVISION OF RADICALS OF THE SECOND DEGREE. 1st. Find the quotient of the parts under the radical, and place it under the common radical. 2d. If the radicals have coefficients, divide the coefficient of the dividend by that of the divisor, and prefix the result to the common radical. NOTE.-When a radical quantity has no coefficient prefixed, its coëffi. cient is understood to be 1. Thus, √2 is the same as 1/2. See Art. 32. ART. 204. To reduce a fraction whose denominator is either a monomial or a binomial containing radicals of the second degree, to an equivalent fraction having a rational denominator. REVIEW.-203. What is the rule for the division of radicals of the second degree? On what principle does it depend? |