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Approximate Roots of Equation.

3. Find the real root of the equation
23. 12 202 + 57 x - 94 = 0.

Ans. x = 3.36216.

331. Corollary. If the given equation is a binomial one, as in art. 223, we can obtain, by this process, a root of any degree whatever.

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1. Extract the square root of 5 by means of continued fractions. Solution. Representing this root by x, we have

m2 = 5, whence

A = 2;

and the substitution of

=

x' gives

2 - 4 -1=0; whence we have

a = 4; and the substitution of

1 a' = 4+

2011 gives

x112 - 4 x!! which, being precisely the same with the equation form, we may conclude that

4 =a= a' = a = al = &c.

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Approximate Roots of Equation.

and the approximating values are

21, 217, 247, 245, &c.; and the value in decimals is

2.23606.

2. Extract the third root of 46 by means of continued fractions.

Ans. 3.58305.

3. Extract the third root of 35 by means of continued fractions.

Ans. 3.271.

4. Extract the square root of 2 by means of continued fractions.

Ans. 1.4142136.

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EXPONENTIAL EQUATIONS

AND

LOGARITHMS.

EXPONENTIAL EQUATIONS

AND

LOGARITHMS.

SECTION I.

EXPONENTIAL EQUATIONS.

1. An Exponential Equation is one in which the unknown quantity occurs as an exponent.

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Solution. This equation is readily solved by means of continued fractions, as explained in Alg. art. 324.

EXAMPLES.

1. Solve the equation

3x = 100.

Solution. Since we have

34 = 81, and

35 = 243,

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