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16y? 8.x Example. Find the square root of 24+

x2

y Arrange the expression in descending powers of y. 16y: 327 8.0 x2

4y
+ 24 -

4+
22
1672
22

32y

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+24

C

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+

y

8.0
8 +

y yo Here the second term in the root, - 4, arises from division of By

8y and the third term, arises from division of 8 by

; 2

y' thus 8;

-8x
Ey Y

EXAMPLES XXIII. C.

32y

by

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2xy

x2

+

22

+4.

323

9.x2

Find the square root of each of the following expressions: x2

4.xC x2 1. - 3x +9.

2. 4

+ 4

y X2

10x . + +ya

4.

+25. 25 5

X2 2.x

2ax

a? 5.

6.

+ 4y2 y

y2 by 72• 64.72

25 7. + +4.

8. - 2 +
9yo 3y

25
af
a3

1 9.

+
-a+1.

10. 24+2.203
64 8
25
67

29 11.

1 - 3a3+

a?. 12. 24 – 2x + + 9

22 – 6.03.

g 3 a4

322

1
13.
+

14. 24 - 2.33+
4
X2

+
2

169 ar2 а2 4ax 15. +4x2+

- 2.23 3

9

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To FIND THE CUBE Root OF A COMPOUND EXPRESSION.

183. Since the cube of a +b is a3 + 3a-b+ 3ab2 +68, we have to discover a process by which a and b, the terms of the root, can be found when a3 +3a2b+3ab2 +63 is given.

The first term a is the cube root of a?.

Arrange the terms according to powers of one letter a; then the first term is a, and its cube root a. Set this down as the first term of the required root. Subtract a3 from the given expression and the remainder is

3a6 +3ab2 + b3 or (3a + 3ab+62) xb. Thus b, the second term of the root, will be the quotient when the remainder is divided by 3a2 +3ab+62.

This divisor consists of three terms:

1. Three times the square of a, the term of the root already found.

2. Three times the product of this first term a, and the new term b.

3. The square of b.
The work may be arranged as follows:

a +3aPb+3al2 +13 (a+b

a3
3(a) =3a2

3a25+3ab2 +33
3x axb=
(6)

3a2+3ab+12 3a2b+ 3ab2 +63
Example 1. Find the cube root of 8x3 – 36x2y + 54.xy27y3.

8.x3 – 36x2y + 54xya 27y3 (2x 3y

8.03 3 (2x) =12.x?

- 36x2y + 54xy 2743
3 x 2x x ( - 3y)= - 18xy
(-3y)=

+9yo
12x2 - 18xy +"ya - 36xạy + 54xy 27y3

+3ab

+62

Example 2. Find the cube root of 27 +108x + 90x2 – 80.43 – 60x4 +48x5 - 8x6.

27 +108x + 90x2 – 80.x3 – 60x4 + 48.05 – 8.36 (3 + 4x – 2x2

27
3 x (3)
=27

108.2 + 90x" – 80x3
3 x 3 x 4.r = + 36.x
(4x)2=

+16.x2

27 +36x + 16.x2 108.2 + 144.x2 + 64.23
3 x (3+4.c)2
=27+72x + 48.r2

54.x” – 144.23 – 60.x4 + 48.25 – 8.26
3 (3+4x) x ( - 2x^)= - 18x” – 24.x3
( – 2x2)

+ 4x4
27+72x + 30x2 – 24x3 + 4x4

54x2 – 144.x3 – 60.x4 + 48x5 – 826

Explanation. When we have obtained two terms in the root, 3+4x, we have a remainder

- 54x2 - 144x3 – 60x4 +48.05 – 8x6.
Take 3 times the square of the root already found and place the result, 27+72x +48x2, as the first part
of the new divisor. Divide – 54xạ, the first term of the remainder, by 27, the first term of the divisor; this
gives a new term of the root - 2x2. To complete the divisor we take 3 times the product of (3+4x) and - 2x”,
and also the square of - 2x. Now multiply the complete divisor by - 2x2 and subtract; there is no
remainder and the root is found.

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EXAMPLES XXIII. d.

Find the cube root of each of the following expressions : 1. Q3 +3a2+3a +1.

2. 23+ 6x2 + 12x +8. 3. a°23 – 3a_x_y2 + 3axy4 yo. 4. 8m3 – 12m2 +6m – 1. 5. 6493 – 144a2b+108ab2 2763. 6. 1+ 3x + 6x2 +7.203 +6x4 +3.25 + 206. 7. 1-6x+21x2 – 44.23 +63x4 – 54.25 +27x6. 8. 43 +6a26 3a c+12ab2 – 12abc+ 3ac2 +863 – 126*c+66c2 – 63. 9. 846 – 365 +66a4 – 63a’ +33a? – Sa+1. 10. y - 3y +6y4-773 +672 - 3y +1. 11. 826 + 12.25 – 30.x4 - 35x3+45x2 +27x – 27. 12. 276 – 54.coa+117cao – 116ca3+1172*a* – 54c5+27a8. 13. 2726 – 27.26 – 1824 +17.003 + 6x2 – 3x – 1. 14. 24.x+y2 +96cRyt - 6.06y + 26 - 96xy6 +6476 - 56.2-y3. 15. 216+342x2 + 171x4 +27x6 – 27.25 – 109.x3 – 108x.

184. We add some examples of cube root where fractional terms occur in the given expressions.

Example. Find the cube root of 54 – 27x3 +

8 36

X6x3 Arrange the expression in ascending powers of x.

8 36
26

+54 – 27x3
23
8

3.0

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yo

EXAMPLES XXIII. e.
Find the cube root of each of the following expressions :
3.x2 3.0

23 1.

222 + -1.

2.

+

+4x+8. 8 4 2

27 3 2

27.23 27x2 93 3. 843 – 4.x?y2 +õryt

4.

+ 8. 27'

64y3 Hy2 + y 27 27

206 5. 23 - 9x +

6. -6.74 +12.cy3 - 8y. 2003

9x 7.

9y, 6y2
4.
y

18 27 27 8.

+ 2x – 7+
27
23 12.x2
54x

1080 48a2 8a3 9.

+

- 112 +
a3
a

X

x3

6x2

33 33

+

+

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23

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3

x2
+

203

+

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-7+

ta

a

27+

66 βα a3 3a? 372 73 11. +

a b 13 79

60.x4 80.23 90x2 8x6 108x 48.25 12.

+ +
y4 y3 ya

yo
y

у° 185. The fourth root of an expression is obtained by extracting the square root of the square root of the expression.

Similarly by successive applications of the rule for finding the square root, we may find the eighth, sixteenth...root. The sixth root of an expression is found by taking the cube root of the square root, or the square root of the cube root.

Similarly by combining the two processes for extraction of cube and square roots, certain other higher roots may be obtained.

Example 1. Find the fourth root of

81x4 - 216x’y + 216.xoy- 96xy3 +1644. Extracting the square root by the rule we obtain 9x2 - 12xy + 4yo; and by inspection, the square root of this is 3x – 2y, which is the required fourth root.

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