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

8y

and the third term, arises from division of 8 by

y

EXAMPLES XXIII. c.

Find the square root of each of the following expressions:

15.

+4x2+ +

2x3.

TO FIND THE CUBE ROOT OF A COMPOUND EXPRESSION.

183. Since the cube of a+b is a3+3a2b+3ab2+b3, we have to discover a process by which a and b, the terms of the root, can be found when a3+3a2b+3ab2+b3 is given.

The first term a is the cube root of a3.

Arrange the terms according to powers of one letter a; then the first term is a3, 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

3a2b+3ab2+b3 or (3a2+3ab+b2) × b.

Thus b, the second term of the root, will be the quotient when the remainder is divided by 3a2+3ab+b2.

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.

Example 1. Find the cube root of 8.x3 – 36x2y+54xy2 – 27y3.

Example 2.

Find the cube root of 27 +108x + 90x2 - 80x3 - 60x1+48x5 – 8x6.

27+108x+ 90x2- 80x3- 60x4+ 48x5 – 8x6 (3 + 4x − 2x2

27

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

-54x2-144x3- 60x4+48x5 - 8x6.

2x2,

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 - 54x2, 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
and also the square of -2x2. Now multiply the complete divisor by -2x2 and subtract; there is no
remainder and the root is found.

1.

EXAMPLES XXIII. d.

Find the cube root of each of the following expressions:

3. a3x3-3a2x2y2+3axy1 — yo. 4.

5. 64a3-144a2b+108ab2 – 2763.

6. 1+3x+6x2+7x3+6x2+3x5+x6.

x3+6x2+12x+8.
8m3-12m2+6m-1.

7. 1-6x+21.x2 - 44x3+63xa – 54x5 +27x6.

8. a3+6a2b-3a c+12ab2-12abc+3ac2+863 – 126°c+6bc2 – c3.

9.

8a6-36a5+66a1 – 63a3 +33a2 - 9a+1.

10. y6-3y+6y4-7y3+6y2-3y+1.

11. 8x+12x5 – 30x1 – 35x3 +45x2+27x − 27.

12.

27x6-54x5a+117x4a2-116x3α3+117x2a1 — 54xα3 +27α®.

13. 27x6-27x5 – 18x4+17x3+6x2 - 3x-1.

14. 24x4y2+96x2y1 — 6.x3y+x3-96xy5+64y6-56x3y3.

15.

216+342x2+171x2 + 27x6 — 27x5 - 109x3 – 108x.

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

8

Example. Find the cube root of 54 - 27x3+z6

Arrange the expression in ascending powers of x.

EXAMPLES XXIII. e.

Find the cube root of each of the following expressions:

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

81x4-216x3y+216x2y2 - 96xy3 +16y*.

Extracting the square root by the rule we obtain 9x2 - 12xy +4y2; and by inspection, the square root of this is 3x - 2y,

which is the required fourth root.