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24. Here, in the two foregoing examples, the quotients observe a certain law, which, if attended to, will enable us, after obtaining a few terms, to extend the quotient to any length without dividing further. Thus the first quotient is itata? tai ta*, &c. Now, if we observe, that in each term the power of a encreases by unity, we may continue to add to the former quotient tas tas ta’ta ta'; and so on to ingnity.

The quotient, in the second example, may also be extended, by observing that the powers of the numerators encrease in the series of the odd numbers, and those of the denominators in the series of the eyen numbers.

25. To divide fractions, multiply the numerator of the diyisor by the denominator of thç dividend for the denominator of the quotient, and multiply the denominator of the divisor by the numerator of the dividend for the numerator of the quotient. If one of them be a whole quantity, it may be brought into the form of a fraction by placing 1 for its denominator.

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26. Involution is performed by the succeslive multiplication of any quantity into itself. A quantity multiplied into itself produces the square of the fame; and the square multiplied again by the original quantity produces the cube of the same; and that cube again multiplied by the root gives the biquadratic power. And so on.

27. Simple quantities are involved by multiplying their exponents by that of the power, and prefixing a like power of the coefficient. Thus, the square of b is b2; the cube of 8a is 51203.

28. Positive roots give positive powers; but negative roots give positive and negative powers by turns.

29. Any two quantities connected by the sign +, are called a Binomial ; but if connected by the fign-, a Residual.

EXAMPLE I.

Required the square, cube, biquadratic, sursolid, and fixth

power of a+6.

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as +5a4b4-1023b2+10a263 +5264 +65 =sursolid.

a to

ao +595 ha -10a4!? +102313-+ 5a2b4 + abs

asb +50*12+102363 + 101264 +5abs +6

a' tbas b4-15a4b2n+202333-7-150264 +6abs +bo=6th power.

EXAMPLE

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EXAMPLE II.

equired the square, cube, and biquadratic powers of a-ob.

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30. It appears by reviewing these examples, that all the terms of the powers of a binomial are positive ; but the terms of the powers of a residual are positive and negative alternately, the first positive, the second négative; the third positive, the fourth negative, and so on, + and by turns. Also, that the sum of the exponents of a and b, in any of the intermediate terms, is equal to the exponent of the first or last term; and that the exponent of the first or of the last term is equal to that of the power. In the first term b is wanting, and the power of a in every succeeding term decreases regularly by 1; and that of b encreases in each term by 1, until a dirappear.

The coefficient of the first term is 1 : The coefficient of the second term is equal to the exponent of the first. One or more terms being found, the cofficient of the next fucceeding term may be discovered in this manner : Multiply the exponent of a in the last term by the coefficient of the fame ; divide the product by the number of terms already made up, and the quot will be the coefficient required.

31. From these observations we may infer the following rule, commonly called the a+b, or the Binomial Theorem, by which we may involve either a binomial or a residual root to a power of any dimension.

RULE

ist, To find the first term of the power.

Multiply the exponent of a in the root by that of the power, for the first term of the power required.

2d, To find the second term of the power.

Multiply the exponent of a in the first term by the coefficient of the same, divide the product by the number of terms already found : the quotient will be the coefficient of the second term of the power; then diminish the exponent of

a,

and encrease the exponent of b, each by 1, for the second term.

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3d, To find the third term of the power. Multiply the exponent of a in the second term by the coefficient of the fame divide the product by 2, (the number of terms already found); the quotient gives the coefficient of the third term; then decrease the exponent of a in the second term, and encrease that of " in the same, each by unity, for the third term.

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