√54√3a=√27×2√3a

= † 27 × √ 2√3a= 3√ 2√3a

=

= 3√ √ 12a=3√12a

which is not a perfect cube, is composed of the factors 27 and 2, the former of which, 27, is a perfect cube. We take the cube root of 27, which is 3. We square the 2, since it is not a perfect cube, and introduce it as factor under the sign. The 12a under the sign is not a perfect square; hence, we denote its root by multiplying the index of the sign by the index of the required root. We have then for the entire root 312a.

281. Rule for the Evolution of Radicals.-Extract the required root of the rational and radical parts, if possible, and reduce the result to its simplest form.

If the rational part is not a perfect power, introduce it under the radical sign; and if the radical part is not a perfect power, multiply its index by the index of the required root.

When the quantity to be evolved is affected by a fractional exponent, the evolution may be performed by dividing the exponent of each letter by the index of the required root.

This transformation is often of great utility, especially in finding the numerical values of fractional radicals.

CASE I.

Rationalization of any Monomial Surd.

283.—Ex. 1. Rationalize √a and a3.

SOLUTION. Multiplying √a by √α, we have a, which is a rational quantity.

Multiplying a by a3 (which is the same quan

tity, with such a fractional exponent as will make the sum of the fractional exponents equal to 1), we have a, which is a rational quantity.

2. What factor will rationalize

?

Ans. a

284. Rule for the Rationalization of any Monomial Surd.— Multiply the surd by the same quantity with such a fractional exponent as, when added to the given exponent, shall be equal to one.

CASE II.

Rationalization of Binomial Surds of the Second Degree.

285.-Ex. 1. Rationalize √/a+√/b.

SOLUTION. The product of the sum and the difference of two quantities is equal to the difference of their squares; hence, va+vb multiplied by va-vb is equal to a-b, which is rational.

ya-vb a-b

2. What factor will rationalize √α-√/b? Ans. √/a+√/b.

286. Rule for Rationalization of Binomial Surds of the Second Degree. Multiply the binomial by the same expression with the sign of one of the terms changed.

PROBLEMS.

1. What factor will rationalize a+3/8? Ans. a-3√/8. 2. Rationalize √7-√5.

3. What factor will rationalize 5-√a? Ans. √5+√α. 4. Rationalize 5+1/3.

CASE III.

Rationalization of either Term of a Fraction.

α

287.-Ex. 1. Reduce to a fraction whose denominator is rational.

288. Rule for the Rationalization of either Term of a Fraction. Multiply both terms of the fraction by such a factor as will render either term rational, as may be required.

RADICAL EQUATIONS SOLVED LIKE SIMPLE EQUATIONS.

289. Radical Equations are those which contain the unknown quantity under the radical sign.

In the process of solving such equations, it will be found necessary first to rationalize the surd expression of the unknown quantity, and then to find its value.

290.-Ex. 1. Find the value of x in √x+4= 9.

2. Find the value of x in √x+9=x+1\

291. Rule for Solving Radical Equations.-Transpose the terms, so that the radical part shall stand as one side of the equation; then involve each side to a power of the same degree as the radical. If there be still a radical part, transpose and involve as before; and, finally, find the value of the unknown quantity as in ordinary simple equations.

10. Given √x+11− 5 = √ x − 4, to find x.