Ex. 4. 23/5 × 3√/20 × √/10 = 63/5 × √(22 × 5) × √(2 × 5) = 612/5* × 12/(26 × 53) × 12/(26 × 56)

12

=612/(513 × 212) = 6012/5.

When the radicands contain numerical factors it is advisable to express them as powers of the smallest possible bases, as in Ex. 4.

It is frequently desirable to introduce the coefficient of a surd under the radical sign.

Ex. 5. 3a (2 ab) = 3/ (27 a3) × 3/(2 ab) = 3/(54 a1b).

13. Multiplication of Multinomial Surd Numbers. - The work may be arranged as in multiplication of rational multinomials.

Ex. Multiply 2-3√√2+5√6 by √2-√3.

Observe that the terms of each partial product are simplified, and that similar surds are then written in the same column.

14. Conjugate Surds. Two binomial quadratic surds which differ only in the sign of a surd term are called Conjugate Surds. E.g., √3+√2 and −√3+√2; 1−√√5 and 1 + √5. Either of two conjugate surds is the conjugate of the other. The product of two conjugate surds is a rational number.

For, (√a+√b)(√a −√√b) = (√a)2 — (√b)2 = a − b.

15. Type-Forms. Many products are more easily obtained by using the type-forms given in Ch. VI., § 1.

Ex.

(√2 + √3)2 = (√2)2 + 2√2 × √3 + (√3)2
=2+2√6+35+2√6.

Find the value of each of the following expressions, without performing

65. (√a+x+√a-x)2, 66. (√2 a2 + √2a)3. 67. (x+2√x2 – 1)3.

68. (Va+b+√a −√b) (√ a + b − va + √b).

69. [√√(a + b) + √(a−b) −√√(a + b) − √ (a — b)]2.

In each of the following expressions introduce the coefficient under the

16. Division of Monomial Surds. The quotient of one monomial surd divided by another is obtained by applying the principle

If the surds are of different orders, they should first be reduced to equivalent surds of the same order.

17. Division of Multinomial Surd Numbers. -It is better to write the quotient of one multinomial surd number by another as a fraction, and then to simplify this fraction by the method to be given in Art. 26. But if the divisor is a monomial, the work proceeds as follows:

(√72 + √32 −4) ÷ 2√2=√36+ √16

2

2

2 √2
=3+2√2=5-√2.

18. Type-Forms. Many quotients are more easily obtained by using the type-forms given in Ch. VI., § 2.

Ex. (V/a2 — √/b2)÷(Va−V/b)=[(V/a)2 − (V/b)2]÷(Va— 3/b)

=Va+vb.

31. (a√a+b√b)÷(√a+√b). 32. (x√x − y√y) ÷ (√x − Vy).

19. From the identity

we infer:

Surd Factors.

(mx + n)2 = m2x2 + 2 mnx + n2

If a trinomial, arranged to descending powers of a letter, say x, be the square of a binomial, the third term is equal to the square of the quotient obtained by dividing the coefficient of x by twice the square root of the coefficient of x; that is,

Consequently, if to any binomial of the form m2x2 + 2 mnx

= n2, be added, the resulting trinomial will

be the square of a binomial.

This step is called completing the square.

E.g., if to 9x+5x we add (2), = }} 9x2+5x+33, =(3x+8)2.

we have

20. An expression of the second degree in a letter of arrangement, say x, can be transformed into the difference of two squares, and hence be factored.

Ex. Factor 25 x2 + 13 x + 1. To transform 25 x2+13 squares, we first complete 25 x2 + 13 x to the square of a binomial by adding (2135), = 168; and, in order that the value of the given expression may remain unchanged, we also subtract from it. We then have

+1 into the difference of two

100

25 x2+13x+1 = 25 x2 + 13x+168-168 +1

100

=(5x+}) _()

=

10

100

· (5 x + 1 3 + √ √69) (5 x + 18 − fu√69).

21. If the coefficient of x2 in the expression to be factored be 1, the term to be added to complete the square is evidently the square of half the coefficient of x.

Ex. 2. Factor-3x2+4 xy + 2 y2.

Since the coefficient of 2 is not the square of a rational number, the work is simplified by first taking out the factor - 3. We then have

-3x2+4 xy + 2 y2=-3(x2-xy — } y2).

Completing xy to the square of a binomial by adding (y)2, y2, to the expression within the parentheses, and also subtracting y2 from it, we obtain

-3x2+4 xy + 2 y2=-3(x2 - xy + y2 - 1o y3)

This method can of course be applied when the factors are rational, but the methods given in Ch. VIII., § 1, Arts. 9-13, are, as a rule, to be preferred.