From the definition of a power, and the rule for multiplication of fractions, we have, 2a2x 2a2x 2a2x3 2a 3by 3by 3by and similarly for other fractions. Hence, the = 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. • 1. (= 24). 4. 4x 2ax2y (20 2 dx 13 6. (- 3/4) 3y2 3 7. (122) 2bz2 8. (-a) X 4 RULE. Ba6x3 2763y3 ; Ans. Ans. a2 b2 16a11 81y' - 27y3 64x3 4a2x4y2 96204 d 33 27y6 also, 2ab2c3a+16 3 3 2 1. (a-2)2. 2. (x-3)-2. 3. (2x2y3)-2. 4. (2x-2y-3)-8, Ans. 101. 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-2x-1, and that power of 2ax2, 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, (3α-2x-4)2 = 1 (2αx2)3 (2αx2)-3= Transferring the factors to the numerator, we have, (3α-2x-4)2 = 9a-4x-8; also, (2αx2) — 3 =2-3α-3x-6 EXAMPLES. Ans. 1 23a3x6 64a6b12c13224 729 In the same way, the truth of the principle may be shown in all similar cases. Ans. a- -4 Ans. xy-2. Ans.xy-6. Ans xy. (x + y)5 BINOMIAL FORMULA. 102. 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 formula. 103. 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. 104. The following powers of x+y are found by actual multiplication: (x + y)1 = x + y. (x + y)2 = x2 + 2xy + y2 (x + y)3 = x3 + 3x2у + 3xy2+ y3. (x + y)1 == x+ + 4x3y + 6x2y2 + 4xy3 + y*. = x + 5x1y + 10x3y2+102y3 + 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 follow ing 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 that letter goes on diminishing by 1 in each term towards the right till the last term, where it is 0: the ponent 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. 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. 2. LAW OF COEFFICIENTS. Let us assume that these laws of formation hold true for a power whose exponent is m, any positive whole number. The application of these laws gives, m+1 If both members of this equation be multiplied by (y), the first member will become (x + y) +1; to find what the second member becomes, let us perform the multiplication. 1 my + m· + m m m m + m = m m. m 2 2 3 (m + 1) m(m − 1). 1.2.3 2 2 (m + 1) m(m 2 1 2 m 3 1 1 · xm− 1y2 + &c. + xym mxm-1y2 + &c. + mxym + ym+? m 1) 2 3 4 (m + 1) m(m − 1) (m − 2). · 1.2.3. 4 2 xm-1y2 + &c.+ mxym + ym+1 + 1 1 +m. m. = m · m The law of the coefficients is evident. (x + y)m +1 = x2+1 + (m + 1)xmy + (m + 1)m 1.2 Substituting these, in the product, we have, (m + 1)me 1.2 2 xm- 2y3 +, &c., + ym+!, + 1) ゾ |