Ex. 3. Find the difference between 5/20 and 345. Here and 5√20=5√4x5=10√5, 3√45=3√9X5= 9√5. Whence the difference = √5. Ex. 4. Find the difference between 250 and 18. Ex. 6. Find the difference between √80ax a... √20a3x3. PROBLEM VI. To multiply surd quantities together. (164.) Let it be required to multiply Va by b. The product will be Vab. For if we raise each of these quantities to the power of n, we obtain the same result, ab; hence these two expressions are equal. We therefore have the following RULE. When the surds have the same index, multiply the quantities under the sign by each other, and prefix the common radicu sign. If there are coefficients, these must be multiplied separately. Ex 1. Required the product of 3√8 and 2√6. Proof. Square 3/8, and we obtain 9×8=72. 72 multiplied by 24=1728. Also, 243 squared =576×3=1728. Ex. 2. Required the product of 5 √8 and 3 √5. Ans. 30 √10. Ex. 3. Required the product of 718 and 54. Ans. 709 Ex. 4. Required the product of 16 and 17. In the preceding examples, let all the results be reduced to their simplest form. If the surds have not the same index, they must first be reduced to a common index, by Art. 161. Ex. 6. Required the product of √2 and 3. (165.) We have seen, in Art. 50, that powers of the same quantity may be multiplied by adding their exponents. The same principle may be extended to roots of the same quantity. Let it be required to multiply ✔a by Va, or a by at. We have seen, in Art. 161, that a2=a, and a3=aa. 1 The product, therefore, is aa×a×a×a×a3=a Hence, roots of the same quantity may be multiplied by adding their fractional exponents. 1 ཝཱ Ex. 1. Multiply 5a by 3a3. Ex. 2. Multiply 3a3 by 21aa. Ans. 15a. Ex. 3. Multiply 3x13 by 4x3y3. Ex. 4. Multiply (a+b)" by (a+b)". (166.) If the rational quantities, instead of being coefficients of the radical quantities, are connected with them by the signs + or -, each term of the multiplier must be multiplied into each term of the multiplicand. by 1. Let it be required to multiply 3+ √5 We obtain the product which reduces to 2- √5 6+25 -3/5-5. 6— √5—5, 1- √5. 2. Multiply 7+2√6 by 9-5√6. 3. Multiply 9+2√10 by 9-2√10. Ans. 3-176. Ans. 41. PROBLEM VII To divide one surd quantity by another. (167.) Let it be required to divide Va' by Va2. The quotient must be a quantity which, multiplied by the divisor, shall produce the dividend; we thus obtain Va; for, according to Art. 164, Vax Va=Va'; Quantities under the same radical sign may be divided like. rational quantities, the quotient being affected with the common radical sign. If there are coefficients, they must be divided separately. If the radicals have not the same index, we must first reduce them to a common index. EXAMPLES. 1. It is required to divide 8108 by 2√6. As the radicals in this last example have not the same index, they must be reduced to a common index. Hence 4\/12=4(12)3=4(12)*=4(144)*. 4(144)*_2('44)*=2('')*=2VY. 2(27)+ (168.) We have seen, in Art. 67, that, in order to divide quantities expressed by the same letter, we must subtract the exponent of the divisor from the exponent of the dividend. The same principle may be extended to fractional exponents. Thus, let it be required to divide a by a3. According to the preceding Article, Hence a root is divided by another root of the same letter or quantity, by subtracting the exponent of the divisor from that of the dividend. (169.) To raise surd quantities to any power. Let it be required to find the square of a3. The square of a quantity is found by multiplying it by itself Again, let it be required to find the cube of a. The cube of a quantity is found by multiplying it by itself twice. Hence the cube of a3 is equal to a3xaxa3=a*; In the same manner we should find the nth power of a==a". RULE. Radical quantities are involved by multiplying their fractional exponents by the exponent of the required power. Ex. 1. Required the fourth power of a3. Ex. 2. Required the cube of v3. Ex. 3. Required the square of 3 V3. Ans. √3. Ex. 4. Required the cube of 17 √21. Ex. 5. Required the fourth power of √6. (170.) If the radical quantities are connected with others by and —, they must be involved by a multiplication of the signs the several terms. Ex. 2. Required the square of 3+2√5. These two examples are comprehended under the rule in Art. 60, that the square of the sum of two quantities is equal to the square of the first, plus twice the product of the first by the second, plus the square of the second. Ex. 3. Required the cube of √x+3√y. Ex. 4. Required the fourth power of √3-2. Ans. 49–20√6. |