Ex. 9. Find the sum of √20a' and √45a2. Ans. 5a 5. Ex. 10. Find the sum of √54a and √96a2. Ans. 7a6. Ex. 11. Find the sum of √32am and √50am3. Ex. 12. Find the sum of √45a2b1x and √125a2b3x. Ans. 9m √2a. Ans. 8ab √5x. Ex. 13. Find the sum of 2/147 and √75. Ans. 19√3. Ex. 14. Find the sum of 72 and 128. Ans. 14√2. Ex. 15. Find the sum of 180 and 405. Ans. 155. PROBLEM VI. To find the Difference of Radical Quantities. (152.) It is evident that the subtraction of radical quantities may be performed in the same manner as addition, except that the signs in the subtrahend are to be changed according to Art. 48. Hence we have the following RULE. When the radicals are similar, subtract their coefficients, and to the difference annex the common radical. But if the radicals are not similar, and can not be made similar, their difference can only be indicated by the sign of subtraction. QUEST.-Give the rule for the subtraction of radical quantities. Ex. 1. Find the difference between 5mva and 2m ✓a. Ans. 3ma. Ex. 2. Find the difference between 7ab2 and 3ab√2. Ans. 4ab√2. Ex. 3. Find the difference between 75 and √27 Ans. 2√3. Ex. 4. Find the difference between 150 and 24. Ans. 36. Ex. 5. Find the difference between 448 and 112. Ans. 4√7. Ex. 6. Find the difference between 520 and 3√45. Ans. √5. Ex. 7. Find the difference between 250 and 18. Ans. 7√2. Ex. 8 Find the difference between √80ax and √20a'x. Ans. 2a2 √5x. Ex. 9. Find the difference between 2√72a2 and ✓162a1. Ans. 3a/2. Ex. 10. Find the difference between √490am and √40am3. Ans. 5m 10a.. PROBLEM VII. (153.) To multiply Radical Quantities together. Let it be required to multiply ✔a by √b. The product, vax√b, will be √ab. For if we raise each of these quantities to the second power we obtain the same result, ab; hence these two expressions are equal. We therefore have the following RULE. Multiply the quantities under the radical signs together, and place the radical sign' over the product. If the radicals have coefficients, these must be multiplied together, and the product placed before the radical sign. Ex. 1. What is the product of 3√8 and 2√6? Ans. 6 √48 which equals 6 √16×3, or 24 √3. Ex. 2. What is the product of 5/8 and 35? Ans. 15/40 or 30/10. Ex. 3. What is the product of 2√3 and 3√5? Ex. 5. What is the product of 5√2 and 3√8? Ex. 7. What is the product of 75 and 5/15? Ex. 8. What is the product of 26 √xy and 5b √xy? Ans. 10b❜xy. Ex. 9. What is the product of 2 √ab and 5 √bc2? Ans. 10 √ a'b'c' or 10abc. QUEST.-Give the rule for multiplying radical quantities Ex. 10. What is the product of 2a √15a and 3a √3a? Ans. 6a2 √45a2 or 18a3 √5. PROBLEM VIII. (154.) To divide one Radical Quantity by another. Let it be required to divide ✔a3 by va3. The quotient must be a quantity which, multiplied by the divisor, shall produce the dividend. We thus obtain ✓a; for, according to Art. 153, Divide one of the quantities under the radical sign by the other, and place the radical sign over the quotient. If the radicals have coefficients, divide the coeffi cient of the dividend by the coefficient of the divisor, and place the quotient before the radical sign. Ex. 1. Divide 8/108 by 26. QUEST.-Give the rule for the division of radical quantities Ex. 5. Divide 15ab √6xy by 5b √2y. Ex. 6. Divide √20x3 by √5x. Ans. 3a √3x. Ans. √4x3 or 2x. Ex. 7. Divide 6a √ 48x1 by 3 √ 4x2. Ans. 2a√12x2 or 4ax √3. Ex. 8. Divide 24 ab2 √12ax by 12ab √3a. Ans. 2b √4x or 4b√x. Ex. 9. Divide 6a' √50x by 3a √5x. Ans. 2a √10x or 2ax2 √10 Ex. 10. Divide 14a'b √72a'b' by 7a√8ab. Ans. 2ab √9ab or 6ab√ab Ex. 11. Divide 6a b'c' √28 by 2a√7. Ans. 3ab'c' √4 or 6ab3c2 Ex. 12. Divide 30a b3 √27a by 15ab √3a. Ans. 2ab9 or 6ab3. PROBLEM IX. To Extract the Square Root of a Polynomial. (155.). In order to discover a method for extracting the square root of a polynomial, let us consider the square of a+b, which is a'+2ab+b'. If we write the terms of the square in such a manner that the powers of one of the letters, as a, may go on continually decreasing, the first term will be the square of the first term of the root; and since, in the present case, the first term of the square is a2, the first term of the root must be a. QUEST. Explain the method of extracting the square root of a poly. nomial. |