MISCELLANEOUS EXERCISES

298. 1. Simplify (V – 8 a-2) 2.

2. Multiply 2x1y1+y ́1 by x − y ̄t.

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3. Divide + 3 x2 + 3 x ̄1 + 1 by x2 + 2 x¬1 + 1.

x-3+3x-2+3x-1+1 x 2 + 2 x 1 +1

Divide:

40. a3+a3b‡+b1 by aa — a‡ba + b3.

41. æ3 +x3 +1 by x3 + x − x3.

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3 x3 + x − 2 x3.

42. 2x-3x-4 + x ̄}
4+x ̄3 by 3x + x
43. a3a32a by 3-2a+4 a ̄3.

44. (a+b+c1) (a1 + b1 — c−1).

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45. 4 a1-5 x1+6 ax2 by a2 ̄1 +3 a3x¬2.

46. m +3 m3

47. x+x
xan

4 m3
m3 by 3–5 m3 — 7 m ̄3.

+1 by x

-3n

58. 8a-"-8a” +5 a3 −3 a¬3 by 5a"-3 a ̃".
59. 16 a-3-6 a2+5 a1+6 by 2 a1+1.
60. √a+Vab+ √b by Va+Vab + Vb.
61. x+y−3 x3y3z3 +z by æ3+y3 +23.

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Find the square root of each of the following expressions:

74. x3 +2x3+1.

75. 9 xy2+12 y−1 + 4 x−1.

76. 9 a2-12 + 10 a2 - 4 a−4 + a−6.

77. 4a-2x2-12 a-1x + 25-24 ax-1 +16 a2x-2.

78. 1+4a-2a-4 a1+25 a ̄-24 a ̄ +16 a-2.

а

Find the cube root of each of the following expressions:

RADICALS AND SURDS

299. A Rational Number is a number expressible either as an integer or as a fraction whose numerator and denominator are integers. The term is seen to include positive and negative integers and fractions.

All other numbers are said to be Irrational.

Thus, 2, §, .333 ... are rational numbers, while √2, (2 + √3)2, √−5

are irrational numbers.

300. Irrational roots of rational numbers (excepting even roots of negative numbers) are called Surds.

Thus, √5 is a surd, since the square root of 5 is irrational; but √2+ √2 is not a surd, since 2 + √2 is not a rational number.

Observe that while every surd is an irrational number there are irrational numbers which are not surds, as 3.1415 ..., √ — 1, etc. It is also to be observed that an algebraic expression that is irrational may or may not be irrational when numerical values are assigned. For instance, vč becomes rational when x = 4.

301. By the introduction of surds we make possible a formal solution of the equation 20, which is satisfied by √2. The surd √2 finds a concrete representation in the diagonal of a square whose side is one unit in length, as shown in the annexed figure.

V 2

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302. Any number or expression which involves a radical symbol (radical sign or fractional exponent) is called a Radical. Thus, √3, √4, 7, Væ, (a + b)2 are radicals. It is seen that radicals may be rational or irrational, but surds are always irrational.