= For, if we raise each expression to the mth power, it becomes Va. Thus, the fourth root the square root of the square root. the sixth root = the square root of the cube root, or the cube root of the = square root. the eighth root the square root of the fourth root, or the fourth root of the square root. the cube root of the cube root. the ninth 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 square root of 64 is 8, and the cube root of 8 is 2. Hence the sixth root of 64 is 2. Ex. 2. Extract the eighth root of 256. Ans. 2. Ans. 37. Ans. 23. Ans. 9. Ans. 15. 218. When the index of a root is the product of two or more factors, and one of the roots can be extracted, while the other can not, a radical may be simplified by extracting one of the roots. To introduce a Factor under the Radical Sign. 219. The square root of the square of a is obviously a, and the cube root of the cube of a is a, etc. That is, Whence, also, a=√a2=√a3=√āa, etc. a√b=√a2x√b=√a2b. Hence, to introduce a factor under the radical sign, we have the following RULE. Raise the factor to a power denoted by the index of the required root, and write it as a factor under the radical sign. EXAMPLES. 1. Reduce ax2 to a radical of the second degree. Ans. Va2x 2. Reduce 2a2bx to a radical of the third degree. Ans. V8ab3x3. 3. Reduce 5+b to a radical of the second degree. To change the Index of a Radical. 220. From Art. 219, it follows that Va=Va2=Va3 =√a1, etc.; Hence we see that the index of any radical may be multiplied by any number, provided we raise the quantity under the radical sign to a power whose exponent is the same number; or the index of any radical may be divided by any number, provided we extract that root of the quantity under the radical sign whose index is the same number. If, instead of the radical sign, we employ fractional exponents, we shall have Hence we see that we may multiply or divide both terms of a fractional exponent by the same number, without changing the value of the expression. 3. 2Va2-2ab+b2 = 2√a−b. 4. 2Va-b=2Va3-3a2b+3ab2—b3. 5. 3V8x3-12x2y+6xy2 —y3 = 3√2x—y. To reduce Radicals to a Common Index. 221. Let it be required to reduce √a and Va to equivalent radicals having a common index. Substituting for the radical signs fractional exponents, the given quantities are Reducing the exponents to a common denominator, the ex pressions are or at and at, Va3 and Va2, which are of the same value as the given quantities, and have a common index 6. Hence 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 new exponent, and take the root denoted by the common denominator. EXAMPLES. 1. Reduce a, a, and a to a common index. Ans. a, a, and a. 2. Reduce a3, a2, and b3 to a common index. 3. Reduce 2, 3, and 5 to a common index. 82, 3 Ans. 'V/64, 81, and 'V125. 4. Reduce 37, 25, and 2 to a common index. 12 Ans. 729, V256, and '512. 5. Reduce a3, a", and a to a common index. 6. Reduce √3, √5, and 7 to a common index. 7. Reduce √2ab, V3ab2, and V5ab3 to a common index. 8. Reduce Va+b, Va-b, and Va2-2 to a common index. To add Radical Quantities together. 222. When the radical quantities are similar, the common radical part may be regarded as the unit, and the coefficient shows how many times this unit is repeated. The sum of the coefficients of the given radicals will then denote how many times this unit is to be repeated in the required sum. If the radicals are not similar they can not be added, because they have no common unit. In such a case, the addition can only be indicated by the algebraic sign. Radicals which are apparently dissimilar may become similar when reduced to their simplest forms. Hence we have the following RULE. Reduce each radical to its simplest form. If the resulting radicals are similar, add their coefficients, and to their sum annex the common radical. If they are dissimilar, connect them by the sign of addition. EXAMPLES. 1. Find the sum of √√27, √48, and √75. 3. Find the sum of √72, 128, and √162. 9. Find the sum of √45c3, √80c3, and √5a2c. 10. Find the sum of √18a5b3+ √50a3b3. 11. Find the sum of Ans. 12√3. Ans. 51 √3. Ans. 23 √5. Ans. √10. Ans. 12V/4. Ans. 9V/5. Ans. √15. Ans. (a+7c) √5c. 12. Find the sum of √4a3b, √25ab3, and 5b √ab. Ans. (2a+10b) Vab. |