the denominator can be made a perfect power by multiplying both terms of the fraction by the necessary rationaliz Multiply both terms of the fraction by a factor that will make the denominator a perfect power of the same degree as the radical. 119. This operation is the reverse of simplifying a radical. Instead of extracting the indicated root of a factor and taking it outside the radical sign, the factor, which is called the coefficient, is to be placed under the radical sign. Since the value of the expression must be unchanged, the factor must be raised to a power of the same degree as the radical. Thus, 2√5= √23 × √5 = √23×5 = √40. Also, 2a√3b= √(2a) 1 × √3b = √16a4X3b48ab. Hence, the following rule: Raise the coefficient to a power of the same degree as the radical and place this power under the radical sign as a surd factor. REDUCTION OF RADICALS TO A COMMON INDEX 120. If radicals are expressed by the use of fractional exponents, the denominator represents the index of the root. If the fractional exponents are changed to fractions having a common denominator, the common denominator will represent the common index of the radicals. 1. Reduce to a common index, Write the radicals with fractional erponents. Change these to equivalent fractional exponents having the least common denominator. Write with radical signs, using the common denominator as the common index and the numerators as the exponents of the respective radicals. If surds of different orders are reduced to a common index, they can be compared as to magnitude. Arrange in descending order of magnitude. 9. √2, 3 and √4. 11.√3, 10 13 and 150. 10. 7, 50 and √5. 12. 5, 7 and 15. ADDITION AND SUBTRACTION OF RADICALS = 121. In algebraic addition, like terms only can be combined by algebraically adding their coefficients. Thus, 2x2 + 3x2 4x2 can be added and their sum written as one term (2 + 3 4)x2 x2, but 2x and 3y can only be added by connecting them with the proper sign, as, 2x + 3y, which is an indicated addition. In the same way, similar surds only can be combined in algebraic addition and are treated just as like terms. Thus 3√3 + 2√3 − 4√√3 = – (3 + 2 − 4)√3 = √3 but 3√2 and 2√3 can only be added by connecting them with the + sign, as, 3√2+2√3. Hence, to add radicals, follow the rule: Reduce all surds to their simplest form. Add the coefficients of similar surds and to this sum annex the common surd factor. Dissimilar surds are connected by their proper signs. √9X2 √25 X 2 + √16 x 2 - √9 X 2 = 5√2 +4√2-3√2 = (5+43)√2=6√2. 2. Simplify 2√3 − 3√! +5√§. 2√3 − 3√! + 5√√§ = 2√§ − 3√§% + 5√§ = 9 36 17. 749 + √28 - 3√63 - √25. 19. 3√m1n 5/8n + √n1 + {√27n2. MULTIPLICATION OF RADICALS 122. In simplifying radicals, it was shown that √ab = √a × √b, hence √a × √b = √ab. This is true of any radicals having a common index. If the radicals do not have a common index, they must be expressed with a common index before multiplying. Reduce the radicals to a common index. Multiply the |