5. Required the 4th power of O.

6. It is required to find the nth power of am.

Ans. 36:

PROBLEM IX.

1

Ans, a

To extract the roots of surd quantities.

RULE.*

Divide the index of the given quantity by the index of the root to be extracted; then annex the result to the root of the rational part, and it will give the root required.

EXAMPLES.

3

1. It is required to find the square root of 93.

Therefore 9/3=3×3 is the square root required.

2. It

* The square root of a binomial or residual surd, A+B, or

Thus, the square root of 8+27=1+√7;

And the square root of 3-1/8= √ 2—1 ;

But for the cube, or any higher root, no general rule is given.

2. It is required to find the cube root of 2,

Therefore 2×2 is the cube root required.

AN INFINITE SERIES is formed from a fraction, having a compound denominator, or by extracting the root of a surd quantity; and is such as, being continued, would run on infinitely, in the manner of some decimal fractions.

But by obtaining a few of the first terms, the law of the progression will be manifest, so that the series may be continued without the continuance of the operation, by which the first terms are found.

PROBLEM I.

To reduce fractional quantities to infinite series.

RULE.

Divide the numerator by the denominator; and the operation, continued as far as may be thought necessary, will give the series required.

EXAMPLES.

1−x) x (1+*+*‍+*3+**+, &c.——, and is the

Here it is easy to see how the succeeding terms of the quotient may be obtained without any further division. This law of the series being discovered, the series may be continued to any required extent by the application of it.

=I−x+**—x3-+-**-, &c. the answer.

Here the exponent of x also increases continually by r from the second term of the quotient; but the signs of the terms are alternately + and ---.

3. Reduce

i

* Here we divide c by a, the first term of the divisor, and

the quotient is, by which we multiply a+x, the whole divi

a

sor, and the product is

a

ed from the dividend c, there remains

; this remainder, be

a

CX

ing divided by a, the first term of the divisor, gives

for the 2d

term

329

x Ans. ×: 1++++, &c.

5. Reduce

term of the quotient, by which we also multiply a+x, the divi

The rest of the quotient is found in the same manner; and four terms being obtained, as above, the law of continuation becomes obvious; but a few of the first terms of the series are generally near enough the truth for most purposes.

And in order to have a true series, the greatest term of the di visor, and of the dividend, if it consist of more than one term, must always stand first.

Thus in the last example ; if x bè greater than a, then x must

be the first term of the divisor, and the quotient will be

ac

+

x

a2c
3

a3c

44

c

x + a

+,&c. the true series; but if x be less

than a, then this series is false, and the further it is continued, the more it will diverge from the truth.

For let a 2, c=1 and x=1; then if the division be perform

ed with a, as the first term of the divisor, you will have

a+x

!

But if x be placed first in the divisor, then will

Now it is obvious, that the first series continually converges to the truth; for the first term thereof, viz., exceeds the truth by