That is, the cube of the sum of two quantities is equal to the cube of the first, plus three times the square of the first times the second, plus three times the first times the square of the second, plus the cube of the second.

The cube of the difference of two quantities is equal to the cube of the first, minus three times the square of the first times the second, plus three times the first times the square of the second, minus the cube of the second.

EXAMPLES.

1. Find the cube of a + 2 b.

We have, (a + 2 b)3 = a3 + 3 a2 (2 b) + 3 a(2 b)2 + (2 b)3

a3 + 6a2b+ 12 ab2 + 8 b3, Ans.

2. Find the cube of 23-3y2.

(2 x3)3 — 3 (2 x3)2(3 y2)+3(2 x3) (3 y2)2 — (3 y2)8

= 8 xo — 36 x6y2 + 54 x3ya — 27 yo, Ans.

Cube each of the following:

The cube of a trinomial may be found by the above method, if two of its terms be enclosed in a parenthesis and regarded as a single term.

15. Find the cube of x2 2 x 1.

(x2-2x-1)= [(x2-2x) - 1]3

=(x2-2x)3-3(x2 -2x)2 + 3(x2 - 2x) - 1

= x2-6x+12x4-8x3-3(x-4x3+4x2)+3(x2-2x)-1 = x¤ − 6 x2 + 12 x1-8 x3-3x2 + 12 x3- 12 x2 + 3 x2-6x-1

XIX. EVOLUTION.

189. If an expression when raised to the nth power, n being a positive integer, is equal to another expression, the first expression is said to be the nth Root of the second.

Thus, if a" b, a is the nth root of b.

=

190. Evolution is the process of finding any required root of an expression.

191. The Radical Sign, √, when written before an expression, indicates some root of the expression.

Thus, Va indicates the second, or square root of a;

Va indicates the third, or cube root of a;

Va indicates the fourth root of a; and so on.

The index of a root is the number written over the radical sign to indicate what root of the expression is taken.

If no index is expressed, the index 2 is understood.

The sign, called the double sign, is prefixed to an expression when we wish to indicate that it is either + or -.

193. From § 192 we derive the following rule:

Extract the required root of the absolute value of the numerical coefficient, and divide the exponent of each letter by the index of the required root.

Give to every even root of a positive term the sign ±, and to every odd root of any term the sign of the term itself.

To find any root of a fraction, extract the required root of both numerator and denominator, and divide the first result by the second.

The root of a large number may sometimes be found by resolving it into its prime factors.

22. Find the square root of 254016.

We have, √254016 = √26 × 34 × 72 = 23 × 32 × 7 = 504, Ans.

23. Find the value of $72 × 75 × 135.

SQUARE ROOT OF A POLYNOMIAL.

194. Since (a + b)2 = a2 + 2 ab + b2, we know that the square root of a2+2ab+b2 is a +b.

It is required to find a process by which, when the expression a2+2 ab+b2 is given, its square root may be determined.

a2+2ab+b2 | a + b

a2

2a + b 2 ab + b2

2 ab+b2

The first term of the root, a, is found by taking the square root of the first term of the given expression.

Subtracting the square of a from the given expression, the remainder is 2ab+b2, or (2a + b)b.

If we divide the first term of this remainder by 2 a, that is, by twice the first term of the root, we obtain the second term of the root, b.

Adding this to 2 a, we obtain the complete divisor, 2 a + b. Multiplying this by b, and subtracting the product, 2 ab + b2, the remainder, there are no terms remaining.

From the above process, we derive the following rule:

from

Arrange the expression according to the powers of some letter.

Extract the square root of the first term, write the result as the first term of the root, and subtract its square from the given expression, arranging the remainder in the same order of powers as the given expression.

Divide the first term of the remainder by twice the first term of the root, and add the quotient to the part of the root already found, and also to the trial-divisor.

Multiply the complete divisor by the term of the root last obtained, and subtract the product from the remainder.

If other terms remain, proceed as before, doubling the part of the root already found for the next trial-divisor.

EXAMPLES.

195. 1. Find the square root of 9 x1 — 30 a3x2 + 25 ao.

9x4 - 30 a3x2 + 25 a6 | 3x2 — 5 a3, Ans.

The first term of the root is the square root of 9x4 or 3x2.

Subtracting the square of 3x2, or 9x4, from the given expression, the first term of the remainder is - 30 a3x2.

Dividing this by twice the first term of the root, or 6x2, we obtain the second term of the root, - 5 a3.

Adding this to 6 x2, we have the complete divisor, 6 x2 — 5 a3.

Multiplying this complete divisor by — 5a3, and subtracting the product from the remainder, there is no remainder.

Hence, 3x2-5 a3 is the required square root.

2. Find the square root of

12 ∞ – 22 x3 +1 − 20 x2 + 9 æo + 8 x + 12 x2.