3. When the two numerators, or the two denominators, can be divided by some common quantity, let that be done, and the quotients used instead of the fractions first proposed.

INVOLUTION.

INVOLUTION is the raising of powers from any proposed root; such as finding the square, cube, biquadrate, &c. of any given quantity. The method is as follows;

MULTIPLY the root or given quantity by itself, as many times as there are units in the index less one, and the last product will be the power required.-Or, in literals, multiply the index of the root by the index of the power, and the result will be the power, the same as before,

Note. When the sign of the root is +, all the powers of it will be; but when the sign is -, all the even powers will be +, and all the odd powers; as is evident from multiplication.

* Any power of the product of two or more quantities, is equal to the same power of each of the factors, multiplied together."

And any power of a fraction, is equal to the same power of the numerator, vided by the like power of the denominator.

di

Also, powers or roots of the same quantity, are multiplied by one another, by

adding their exponents; or divided, by subtracting their exponents. 32

Thus a3 xa2 = a3taas. And a3➡a3 or

= @

a.

the cubes, or third powers, of x-a and x+a.

EXAMPLES FOR PRACTICE.

1. Required the cube or third power of 3a2.

2. Required the 4th power of 2aab.

3. Required the 3d power of -4a2b3.

4. To find the biquadrate of

262

5. Required the 5th power of a−2x. 6. To find the 6th power of 2a1.

SIR ISAAC NEWTON'S RULE for raising a Binomial to any Power whatever.*

1. To find the terms without the Co-efficients. The index of the first, or leading quantity, begins with the index of the given power, and in the succeeding terms decreases continually by 1, in every term to the last; and in the 2d or following quantity, the indices of the terms are 0, 1, 2, 3, 4, &c. increasing always by 1. That is, the first term will contain only the first part of the root with the same index, or of

Note. The sum of the co-efficients, in every power, is equal to the number 2, when raised to that power. Thus 1+1=2 in the first power; 1+2+1 =4=2 in the square; 1 +3 +3 +18=23 in the cube, or third power; and so on.

the

the same height as the intended power: and the last term of the series will contain only the 2d part of the given root, when raised also to the same height of the intended power; but all the other or intermediate terms will contain the products of some powers of both the members of the root, in such sort, that the powers or indices of the 1st or leading member will always decrease by 1, while those of the 2d member always increase by 1.

2. To find the Co-efficients. The first co-efficient is always 1, and the second is the same as the index of the intended power; to find the third co-efficient, multiply that of the 2d term by the index of the leading letter in the same term, and divide the product by 2; and so on, that is, multiply the coefficient of the term last found by the index of the leading quantity in that term, and divide the product by the number of terms to that place, and it will give the co-efficient of the term next following; which rule will find all the co-efficients, one after another.

Note. The whole number of terms will be 1 more than the index of the given power: and when both terms of the root are +, all the terms of the power will be +; but if the se. cond term be all the odd terms will be +, and all the even terms which causes the terms to be + and alternately. Also the sum of the two indices, in each term, is always the same number, viz. the index of the required power and counting from the middle of the series, both ways, or towards the right and left, the indices of the two terms are the same figures at equal distances, but mutually changed places. Moreover, the co-efficients are the same numbers at equal distances from the middle of the series, towards the right and left; so by whatever numbers they increase to the middle, by the same in the reverse order they decrease to the end.

EXAMPLES.

1. Let a+x be involved to the 5th power.

will be

The terms without the co-efficients, by the 1st rule,

a3, a*x, a3x2, α2x2, ax1, x5,

and the co-efficients, by the 2d rule, will be

5X4 10X3 10X2 5X1

1, 5,

or 1, 5, 10, 10, 5, 115

Therefore the 5th power altogether is as+5a4x+10ax2+10a2x2+5x++x3, 27

But

VOL. 1.

But it is best to set down both the co-efficients and the powers of the letters at once, in one line, without the intermediate lines in the above example, as in the example here below.

2. Let a-x be involved to the 6th power.

The terms with the co-efficients will be
a-6a3x+15a4x2-20a3x3+15a2x2-6αx3 +x®,

3. Required the 4th power of a-x.

Ans. a-4a3x+6a2x2-4ax3+x1.

And thus any other powers may be set down at once, in the same manner; which is the best way.

EVOLUTION.

EVOLUTION is the reverse of Involution, being the method of finding the square root, cube root, &c. of any given quantity whether simple or compound.

CASE I.

To find the Roots of Simple Quantities.

EXTRACT the root of the co-efficient for the numeral part; and divide the index of the letter or letters, by the index of the power, and it will give the root of the literal part; then annex this to the former, for the whole root sought.*

Any even root of an affirmative quantity, may be either + or -: thus the square root of a2 is either+a, ora; because +a+a+a, and -axa+a2 also.

But an odd root of any quantity will have the same sign as the quantity itself: thus the cube root of a 3 is a and the cube root of a 3 is -a; for +a×+ ax +a+a3, and -ax-aX-aa3.

Any even root of a negative quantity is impossible: for neither +ax +a, norax can produce -a2.

Any root of a product, is equal to the like root of each of the factors multiplied together. And for the root of a fraction, take the root of the numerator, and the root of the denominator.

EXAMPLES.