96. Let it be required to find the third power of

2a2x

3by

From the definition of a power, and the rule for the multiplication of fractions, we have,

=

2a2x 2α2x 2a2x

=

8a6x8

3by 3by 3by 3by 27b sys

and similarly for other fractions; hence, the

RULE.

Raise the numerator to the required power for a new numerator, and the denominator to the required power for a new denominator.

The rule for signs is the same as in the last article.

Extension of the Preceding Rules.

97. The rule for raising a monomial to any power holds true when the exponents of any of the letters are negative, or when the exponent of the required power is negative.

Let it be required to find the square of 3a ̄x ̄, and that power of 2ax, whose exponent is 3: it has been shown that any factor may be changed from the denominator to the numerator, or from the numerator to the denominator, by changing the sign of its exponent (Art. 32); hence,

Transferring factors to the numerator, we have,

(3α-2x-4)2 = 9a-4x-8;

also, (2αx2)-3 = 2-3a-3 x6 = fa3 x 6,

which results conform to preceding rules.

98. A binomial may be raised to any power by the process of continued multiplication, but when the exponent of the power is greater than 2, the operation is greatly abridged by making use of the binomial for

mula.

Definitions.

99. The binomial formula, is a formula by means of which a binomial may be raised to any power, without going through the process of continued multiplication.

Demonstration.

100. The following powers of x+y are found by actual multiplication:

(x + y) 4

(x + y)5 =

x4 + 4x3y + 6x2y2 + 4xy3 + y2.

x + 5x1y + 10x3y2+10x3y3 + 5xy1 + y3.

And in the same way, the higher powers might be obtained. If we examine the powers already deduced, we see that they are all formed according to the following laws:

1°. LAW OF EXPONENTS.-The exponent of the leading letter in the first term is equal to the exponent of the power, and the exponent of that letter goes on diminishing by 1 in each term towards the right till the last term, where it is 0: the exponent of the following letter is 0 in the first term, and the exponent of that letter goes on increasing by 1 in each term towards the right to the last term, where it is equal to the exponent of the power.

2°. LAW OF COEFFICIENTS. The coefficient of the first term is 1; the coefficient of any succeeding term is found by multiplying the coefficient of the preceding term by the exponent of the leading letter in that term, and dividing the product by the number of terms preceding the required term.

Let us assume that these laws of formation hold true for a power whose exponent is m, m being any positive whole number. The application of these laws gives,

If both members of this equation are multiplied by (x + y), the first member of the resulting equation will be (x + y)m+1: to find what the second member will be, let us perform the multiplication, as indicated below: