PROBLEM VIII.

To extract the root of any pure power in numbers.

RULE*.

1. Let m the number whose root is required; r= nearest root which can be found by trial; and n= to the index.

vx(2v+2n−1)+}×(n−1)×(2n-1)

extremely near.

EXAMPLES.

1. Given x2=2; or, which is the same thing, let the square root of 2 be found.

Suppose the root found by trial to be 1.4; then we shall

have m=2, r=1.4, n=2, and v=;

2× 1.96

2-1.96

=98.

* One of the most convenient rules for practice, which has yet been discovered, is the following: (n+1)rn+(n−1)N: (n+1)n+ (n-1)rnr: the true root nearly.

Where it may be observed that N= given number; r nearest root, found by trial; and n index, as before.

And if the second approximation be used, the root will be found 1.41421356236, which is true to the last place of decimals.

=

2. Given x3-500; or let it be required to extract the cube root of 500..

Suppose the root, found by trial, to be 8; then we shall

have m=500, r=8, n=3, and v=

3X512

-12

128;

And, therefore, x=r+:

rx(6v+n+1)

=8.063=

vx(6v+4n)-2

rx(2v+R)

=8

7.93 for the first approximation.

Or x=r+

6072

96389

(2v+2n−1)xv+3×(n−1)x(2n-1)

=7.937005259936 for the second approximation,

which is true to the last place of decimals.

3. Let it be required to find the cube root of 2.

Ans. 1.259921.

4. Required the cube root of 117.

Ans. 4.89097.

5. What is the sursolid, or 5th root, of 125000?

Ans. 10.456389.

6. It is required to find the 7th root of 100000.

Ans. 5.1794746792.

7. It is required to find the 365th root of 1.05.

Ans. 1.00013366.

PROBLEM IX.

To find the root of an exponential equation.

RULE*.

1. Find, by trial, two numbers, as near the true root as possible, and substitute them in the given equation instead of the unknown quantity, marking the errors which arise from each of them.

2. Multiply the difference of the two numbers, found by trial, by the least error, and divide the product by the difference of the errors, when they are alike, and by their sum when they are unlike.

3. Add the quotient, last found, to the number belonging to the least error, when that number is too little, and subtract it when too great, and the result will give the true root nearly.

4. Take this root and the nearest of the former, and, by proceeding in like manner, a root will be had still nearer than before; and so on, to any degree of exactness required.

EXAMPLES.

1. Given x=100, to find the value of x by approximation.

By the nature of logarithms xxlog. x=log. 100=2. And, since x is found by trial to be greater than 3, and less than 4,

Let, therefore, 3.5 and 3.6 be the two supposed values of x.

The rule for solving exponential equations was invented by M. Jean Bernoulli, and published in the Leipsic Acts, 1697.

Then the log. x=log. 3.5.5440680; and xx log.x-1.9042380

2

-.0957620=1st error, too little.

And the log. x=log. 3.6.5563025; and

xxlog.x=2.0026890

Again, suppose x=3.597; then we shall have log.x=.5559404, and

xxlog.x=1.9997176, which, subtracted from 2, gives .0002824, the third error, too little.

2d number 3.600

3d number 3.597

2d error +.0026890

3d error- -.0002824

3.597285=x=root required nearly.

2. Given x=123456789, to determine the value of ..

Ans. x

8.6400268.

OF INDETERMINATE OR UNLIMITED

PROBLEMS.

A problem is said to be indeterminate or unlimited when the equations, expressing the conditions of a ques tion, are less in number than the unknown quantities to be determined.

And though such kind of problems are capable of innumerable answers, yet the results, in whole numbers, are generally limited to some determinate number, and may be obtained as follows.

PROBLEM I.

To find the values of x and y, in the equation ax=by+c; where a, b, and c are given numbers, which admit of no common divisor.

RULE*.

1. Let wh stand for a whole number, and reduce the

equation to x=

2. Make

by+c_dy+f, by throwing all whole num

a

a

bers out of it, till d and ƒ be each less than a.

* This rule is founded on the following obvious principles : That the sum, difference, or product of any two whole numbers is a whole number.

And that if a number measures the whole of any number, and a part of it, it will also measure the remaining part.

N