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4. 60a4b2 + 12ab+ a + 6466 + 160a3b3 + 240a2b4 + 19265

·

5. 729x62916x5y + 4860x1y2 - 2160x3y3 +2160x2y1 — 576xy5 + 64y6.

Find the eight root of:

6. 256 +1024x + 1792x2 + 1792x3 + 1120x1 + 448x5+ 112x6+16x+x8.

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Find the cube root of:

23. 87 to two decimals.

24. .7 to three decimals.

25. 1 + 6d + d + 6d3 + 15d4 + 20d3 + 15d2. 26. 8x + 1 9x 63x3 + 33x2 + 66x4

Find the fourth root of

36x5.

27. a64aab-1 + 6a3b-1

4ab-1√ab-1+b-2

CHAPTER XV

RADICALS

115. Any indicated root of a number or algebraic expression is a radical expression or simply a radical.

As V8, Va, Va2 + b2.

The quantity under the radical sign is the radicand. An indicated root that cannot be found exactly is called a surd.

As √2, V4, Vx + y.

The order of the surd is shown by the index.

Thus,√2 is a surd of the second order or a quadratic surd.

V4 is a surd of the third order or a cubic surd.

V is a surd of the fourth order or a biquadratic surd, and so on. An expression containing both rational and radical factors is called a mixed surd, as 5√2, xy, and the rational factor is called the coefficient of the surd. When a surd has no rational coefficient except 1, it is called an entire surd. As √5, Vx2- y2.

A surd is in its simplest form when the quantity under the radical sign is integral and as small as possible.

Surds are similar if they have the same surd factor when reduced to their simplest forms.

As, √3, 4√3, 1√3.

REDUCTION OF RADICALS

Case I

116. When the radical is a perfect power whose exponent is a factor of the index of the root.

Note that the word simplify still means to perform the indicated operations.

Thus in simplifying a radical of Case I, the indicated operation is performed as far as possible.

Thus, V2 can be performed only partially as x2 is not

a fourth power; but, since the fourth root is the square root of the square root, one of these square roots can be extracted. This is most easily done by using fractional exponents as explained under the Theory of Exponents.

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Similarly 68.xy

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x1 = √x.

28xly

=

21xly

2x

=

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√27a3b3 (33a3b3/ 3 alb 31a1b1

Hence, the following rule:

Divide the exponent of the power by the index of the root. Note-In problems involving radicals, arithmetical numbers should be resolved into their prime factors.

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117. When the radical contains a factor which is a perfect power of the same degree as the radical.

1. √x2y = √x2 × √y = ̧x√y.

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The indicated operation is performed as far as possible by extracting the indicated root of the factor whose root can be extracted.

Hence, the following rule:

Resolve the radical into two factors, one of which is the greatest perfect power of the same degree as the radical.

Ex

tract the indicated root of this factor which then becomes the rational coefficient of the resulting surd.

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Note. The easiest method of resolving a number into its prime factors is the method usually taught in arithmetic.

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118. When the radical is a fraction whose denominator is not a perfect power of the same degree as the radical.

A surd is not in its simplest form unless the quantity under the radical sign is integral. If the denominator is not a perfect power of the same degree as the radical, the indicated root of this denominator cannot be extracted. The value of the fraction is not changed if numerator and denominator are multiplied by the same factor. Hence,

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