act power of the same name with the root; extract its root; and prefix it to the other factor with the radical sign between them. In the expressions 2√aa and 2a√3a, the quantities 2 and 2a placed before the radical sign are called the coefficients of the radical. EXAMPLES. 1. Reduce V56ab to its most simple form. Ans. 2ab7a. 2. Reduce V54a b c to its most simple form. Ans. 3ab2ac. 3. Reduce 48a b c to its most simple form. Ans. 2ab'c √3ac2. 4. Reduce 192abc1 to its most simple form. 5. Reduce V192a'b'c3 to its most simple form. 6. Reduce 9816' to its most simple form. (159.) There is another principle which can frequently be employed to advantage in simplifying radicals. The square of the cube of a is equal to the sixth power of a. For the square of the cube of a is a3Xa3, which equals a3±3=a®. So, also, the fourth power of the cube of a is equal to the twelfth power of a. And, in general, the mth power of the nth power of any quantity is equal to the mnth power of that quantity. That is (a")"=ɑmn. Hence, conversely, The mnth root of any quantity is equal to the mth root of the nth root of that quantity. Thus, the fourth root = the square root of the square root; 66 the sixth root = the square root of the cube root, or the cube root of the square root; 66 the eighth root 66 = the square root of the fourth root, or the fourth root of the square root; 2 the ninth root = the cube root of the cube root. Hence, when the index of a root is the product of two or more factors, we may obtain the root required by extracting in succession the roots denoted by those factors. Ex. 1. Let it be required to extract the sixth root of 64. The sixth root is equal to the cube root of the square root. The square root of 64 is 8, and the cube root of 8 is 2. Hence the sixth root of 64 is 2. Ex. 2. Let it be required to extract the eighth root of 256. The eighth root is equal to the fourth root of the square root; or to the square root of the square root of the square root. The square root of 256 is 16, and the fourth root of 16 is 2. Hence the eighth root of 256 is 2. When one of the roots can be extracted, and the other can not, a radical may be simplified by extracting one of the roots. Thus, the fourth root of 9 is equal to the square root of the square root of 9; that is, the square root of 3, Or, algebraically, √9= √3. Ex. 3. Reduce V4a to its most simple form. Ans. 2a. Ex. 4. Reduce V36ab to its most simple form. PROBLEM II. (160.) To reduce a rational quantity to the form of a surd. The square root of the square of a is obviously a; that is, So, also, the cube root of the cube of a is a ; Hence, to reduce a rational quantity to the form of a surd, we have the following RULE. Raise the quantity to a power of the same name with the given root, and then apply the corresponding radical sign. EXAMPLES. 1. Reduce 3 to the form of the square root. Here 3×3=3'=9; whence 3=√9. Ans. 2. Reduce ax to the form of the square root. 3. Reduce 2x2 to the form of the cube root. Ans. 8x. 4. Reduce 5+b to the form of the square root. 7. Reduce ab to the form of the square root. 8. Reduce a to the form of the nth root. It will be observed, that this Problem is nearly the reverse of the preceding, and, consequently, brings quantities into a less simple form; nevertheless, this form is sometimes better suited to subsequent operations, as will be seen hereafter. PROBLEM III. (161.) To reduce surds which have different indices to others of the same value having a common index. 1 Ex. 1. Reduce a and a to surds having the same radical sign. From the preceding Article, it is obvious that the square root of a is equal to the sixth root of the cube of a; which are of the same value, and have the common index 6. Ex. 2. Reduce 32 and 23 to a common index. Hence 27 and 4 are the quantities required. Whence we derive the following RULE. Reduce the fractional exponents to a common denominator; raise each quantity to the power denoted by the numerator of its reduced exponent; and take the root denoted by the common denominator.. Ex. 3. Reduce and 4 to a common index. Ans. 4 and 8. Ex. 4. Reduce a' and a2 to a common index. Ex. 5. Reduce and b3 to a common index. 2 Ex. 6. Reduce 53 and 7 to a common index. 1 Ex. 7. Reduce a and b to a common index. PROBLEM IV. To add surd quantities together. (162.) Two radicals are similar when they have the same index, and the same quantity under the radical sign. Thus, 3 va and 5 va are similar radicals. So, also, 7b and 10b are similar radicals. But va and a are not similar radicals; for, although they have the same quantity under the radical sign, they have not the same index. Ex. 1. Find the sum of 2√a and 3√a. As these are similar radicals, we may unite their coefficients by the usual rule; for it is evident that twice the square root of a and three times the square root of a make five times the square root of a. Hence the following RULE. When the radicals are similar, add the coefficients, and annex the radical part. But if the quantities are dissimilar, and can not be made similar by the reductions in the preceding Articles, they can only be connected together by the sign of addition. Ex. 2. Add 16 to 26. Ans. 3 √6. Ex. 3 Add 5Va and -2 Va. Ex. 4. Add a √b+c and x√b+c. If the radical parts are originally different, they must, if pos sible, be made alike by the preceding methods. (163.) It is evident that the subtraction of surd quantities may be performed in the same manner as addition, except that the signs in the subtrahend are to be changed according to Art. 43. Ex. 1. Required to find the difference between 448 and ✓112. Ex. 2. Find the difference between V192 and 24. |