(205.) Formula for the Development of the Binomial (a+b)".

When the exponent n is a positive integer, as 2, 3, or 4, &c., the development will terminate at the term in which the exponent of b becomes equal to the exponent of the Binomial.

For the exponent of a in that term will be 0, and 0 will thus become a factor in finding the coefficient of the next term; hence the next term will be 0, (43).

EXAMPLE I.

To find the 4th power of (a+b), by the Binomial Theorem; that is, to develop (a+b)*.

The literal factors without the coefficients, will be

a4 a3b a2b2 ab3 b4

By computing and introducing the coefficients, with the signs, we a4+4a3b+6a2b2+4•ab3+ba.

have

The student will readily perceive the application of the preceding principles to the several terms of this Power.

Demonstration of the Binomial Theorem.

The principles which have been given for the development of (a+b)", will be demonstrated under the supposition that the exponent n is a positive integer.-A general demonstration would be equally applicable to negative or fractional exponents; such demonstration is unnecessary here, and is too abstruse for the present stage of our subject.

If we multiply together the binomial factors

a+b, a+c, a+d, a+e,

and decompose the product terms which contain the lower powers of a, the final Product may be represented as follows;

In this Product observe that the coefficients of the powers of a in the successive terms, are as follows;

The Coefficient of a4 is 1; the coefficient of a3 is the sum of the second terms, b, c, &c., of the binomial factors composing the Product; the coefficient of a2 is the sum of the products bc, bd, &c., of the second terms of the binomial factors combined two and two, (196); and the coefficient of a is the sum of the products bcd, bce, &c., of the second terms of the binomial factors combined three and three.

Suppose now that c, d, e, &c. are each equal to b, and the number of binomials equal to n. The Product of these factors will be the nth

power of (a+b); that is, it will be

the development of (a+b)".

The Exponent of a in the first term of the Product will evidently ben; and, as exemplified in the preceding multiplication, this exponent will decrease by 1, continually, in the succeeding terms.

Hence we shall have

an-1,

an-2, &c. in the consecutive terms.

The Coefficient of a" will evidently be 1. The coefficient of a"-1 in the second term, will be n times b.

Hence the second term will be

nban-1 or nan-1b.

The Coefficient of a 2 in the third term, will be b2 taken as many times as there are combinations of two letters in n letters, (196). Hence the third term will be

The Coefficient of a"-3 in the fourth term, will be 63 taken as many times as there are combinations of three letters in n letters. Hence the fourth term will be

The several terms thus obtained agree with Formula (205); and the law of development thus indicated, will, in like manner, be found applicable to any number of terms.

With regard to the Signs,-it is evident that when a and b are positive, all the terms in any power of (a+b) will be positive, (42).

When b is negative, all the odd powers of b will be negative, (202); and these negative powers multiplied by the positive powers of a, will cause the 2d, 4th, 6th, &c. terms to become negative.

EXAMPLE II.

To find the 5th power of the binomial a―x.

The literal factors without the coefficients are

a3, a1x, a3x2, a2x3, ax1, x5.

By computing and inserting the coefficients, with the signs, we have

a5-5a4x+10a3x2-10a2x3+5ax1—x5.

By applying the principles of the Binomial Theorem, we obtain

a3+3a2.2b+3a(2b)2+(2b)3 ;

which may be developed into 4. Find the square of 3a+y. 5. Find the cube of 2a-3x.

6. Find the square of a+x-y,

Ans. a3+6a2b+12ab2+8b3.
Ans. 9a2+6ay+y2.

Ans. 8a3-36a2x+54ax2-27x3.

By operating on (x-y) as if it were a monomial, and applying the Binomial Theorem to a+(x-y), we obtain

a2+2a(x−y)+(x—y)2;

which may be developed into Ans. a2+2ax-2ay+x2-2xy+y2. 7. Find the square of a-b+y. Ans. a2-2ab+2ay+b2—2by+y2. 8. Find the square of a-x-y. Ans. a2-2ax+x2-2ay+2xy+y2. 9. Find the square of a+b-2x.

10. Find the square of a2+x3.

Ans. a2+2ab+b2-4ax-4bx+4x2.

This square, according to the Binomial Theorem, may be indicated thus:

EVOLUTION.

(206.) EVOLUTION consists in extracting any required root of a given quantity, regarded as the corresponding power of the root to be found.

Extracting the square root consists in finding a quantity whose square is equal to a given quantity; extracting the cube root consists in finding a quantity whose cube is equal to a given quantity; and so

on..

Roots of Unity, Monomials, Fractions, &c.

(207.) Every root of unity is unity, since the square, or cube, &c. of 1 is 1; thus 1x1=1; 1x1x1=1; and so on.

In general terms, any power or root of a unit is a unit.

(208.) Any required Root of a monomial will be found by extracting the root of its numerical coefficient, and dividing the exponents of its literal factors by the integer corresponding to the root.

Thus the square root of 25 a2x is 5ax, found by extracting the square root of 25, and dividing the exponents of a and x by 2.

And the cube root of 27ax is 3a2x, found by extracting the cube root of 27, and dividing the exponents of a and x by 3.

The correctness of this method will appear from considering that the Extracting of a Root is the reverse of raising the corresponding Power, (200).

Hence also the propriety of denoting roots by fractional expo

nents.

denotes the square root of a, because

axa+=a.

(209.) Any required Root of a fraction will be found by extracting the root of its numerator and denominator, separately.

9a2y4 3ay 2

is

found by extracting the

square root of the numerator, and the square root of the denominator.

A Root of a mixed quantity would be found by extracting the root of the equivalent improper fraction.

(210.) A Root whose exponent is resolvable into two factors, may be found by extracting in succession the roots denoted by those factors.

The 4th root may be found by extracting the square root of the square root,—the exponent which denotes the 4th root, being×1.

Thus the 4th root of x 81 is the square root of 9; root of 100.

is the square root of x2; the 4th root of and the 4th root of 10000 is the square

The 6th root may be found by extracting the square root of the cube root, or the cube root of the square root, being equal to

.

The correctness of this method of extracting Roots, is evident from considering, that, by raising in succession the powers denoted by two or more factors, we shall obtain the power denoted by the product of those factors.

Thus the square of the cube of a is (a3)2=a6, (200), and therefore, conversely, the square root of the cube root is the sixth root

Roots of Powers or Powers of Roots.

(211.) The Numerator of a fractional exponent denotes a power of the quantity affected, and the Denominator a root of that power.

Thus

až

denotes the square root of the first power of a, or simply the square root of a. In like manner a denotes the cube root of the square of a; a denotes the 4th root of the cube of a; and so on. But, observe that

(212.) A Root of any power of a quantity, is equal to the same power of the same root of the quantity.

To illustrate this principle it may be shown that the cube root of the 6th power of a, is equal to the 6th power of the cube root of a.

The cube root of a6 is a2 since a2.a2.a2=a6; and the 6th power of a is also a2, since the cube of a3

is also a2, since the cube of a3 is a, and the square of this cube. —which gives the 6th power of at,—is a2, (200).

To give an application of this principle to numbers, the cube root of the square of 8, is 4, and the square of the cube root of 8 is 4