Multiply:

Exercise 80

1. x2+3x-1-2 by x-2-2x-1-4.

2. a*+2a‡b‡+ba by aa — 2 a‡ba +6$.

3. 4x-3+6x-2m-5x-m-3 by 3x-2m+2x-1.

4. a9a+27 a ̄-27 a ̄ by a—6+9a ̄.

6. c-c-3-8c-2+11 c-1-3 by c-2+2c-1-3.

7. 2a−aa+4a3+4a — 3 by a — aa +3.

8. 35+4a--16a-2m+19a-sm-6a-4m by 7+5a-m-3a-2m. 9. x2+2x‡ −7x1—8x+12x-3 by x − 3 x ̄1+2x1.

Simplify :

11. [(x+3)(x−3)−1— (x−3)(x+3)-1]÷[1−(x2+9)(x+3)−2].

20. x ̄‡—4x ̄13y3+8x ̄3 y§ — 8 x ̄*y* + 4 y3.

Find the value of x in each of the following:

39. x1y, and y2 = 9.

=

40. x=y-1, and y2 = 3.

41. x-3=y, and y-2 = 2. Simplify:

42. x2 = y ̄1, and y = 3.

43. x1=y3, and y3 — 9.

44.

=

CHAPTER XX

RADICALS. IMAGINARY NUMBERS. REVIEW

252. A radical expression is an indicated root of a number or expression.

Thus: √2, 7, √10, and √x + 1 are radical expressions.

253. Any expression in the form V is a radical expression, or radical. The number indicating the required root is the index of the radical, and the quantity under the radical is the radicand.

In 17, the index is 3, and the radicand, 7.

254. A surd is an indicated root that cannot be exactly obtained.

255. A radical is rational if its root can be exactly obtained, irrational if its root cannot be exactly obtained.

Thus: √25 is a rational expression; √10 is an irrational expression.

256. A mixed surd is an indicated product of a rational factor and a surd factor.

Thus: 3√5, 4√7x, ab√a + b are mixed surds.

257. In a mixed surd the rational factor is the coefficient of the surd.

Thus In 4√5x, 4 is the coefficient of the surd.

258. A surd having no rational factor greater than 1 is an entire surd.

Thus: √5 ac is an entire surd.

259. The order of a surd is denoted by the index of the required root.

Thus: √5 is a surd of the second order, or a quadratic surd.

37 is a surd of the third order, or a cubic surd.

260. The principal root.

Since (+ a)2 = + a2 and (− a)2 = + a2, we have √+a2= ±a. That is, any positive perfect square has two roots, one + and the other, but in elementary algebra only the + value, or principal root, is considered in even roots.

THE TRANSFORMATION OF RADICALS

TO REDUCE A RADICAL TO ITS SIMPLEST FORM

261. A surd is considered to be in its simplest form when the radicand is an integral expression having no factor whose power is the same as the given index. There are three common cases of reduction of surds.

(a) When a given radicand is a power whose exponent has a factor in common with the given index.

Hence, to reduce a radical to a radical of simpler index:

262. Divide the exponents of the factors of the radicand by the index of the radical, and write the result with the radical sign. Illustration:

√8 ax3 = √/23ax3 = 2}a‡x} = √/2 a2x. Result.

(b) When a given radicand has a factor that is a perfect power whose exponent is of the same degree as the index.

By Art. 244,

√a3b = (a2b)‡ = a3¿1 = ab‡ = a√b.

Hence, to remove from a radicand a factor of the same power as the given index:

263. Separate the radicand into two factors, one factor the product of powers whose highest exponents are multiples of the given index. Extract the required root of the first factor and write the result as the coefficient of the indicated root of the second factor.

Illustrations:

1. √12 a3 = √√4 a2 × 3 a = 2 a√3 a. Result.

2. 2√72 a3x1y1 = 2√36 a2x1y . 2 ay = 2(6 ax2y3) √2 ay