Ex. 2. Find a multiplier which shall render √3-√x rational, and determine the product.

Ex. 3. Find multipliers which shall render √3-Vx rational, and determine the product.

238. 3d. When the surd is a trinomial.

When a trinomial surd contains only radicals of the second degree, we may reduce it to a binomial surd by multiplying it by the same expression, with the sign of one of the terms changed. Thus, √a+√b+√c multiplied by √a+√b−√c gives for a product a+b-c+2√ab, which may be put under the form of m+2√ab.

Ex. 1. Find multipliers that shall make √5+√3−√2 rational, and determine the product.

Ex. 2. Find multipliers that shall make 1+√2+√3 rational, and determine the product.

To transform a Fraction whose Denominator is a Surd in such a Manner that the Denominator shall be Rational. 239. If we have a radical expression of the form

or

α

α

√ √ it may be transformed into an equivalent expression in which the denominator is rational by multiplying both terms of the fraction by √b±√c. Hence the

RULE.

Multiply both numerator and denominator by a factor which will render the denominator rational.

240. The utility of the preceding transformations will be seen if we attempt to compute the numerical value of a fractional surd.

Ex. 1. Let it be required to find the square root of; that is,

value of the fraction is found to be 0.6546. Ex. 2. Compute the value of the fraction

7√5 √11+√3°

Ex. 3. Compute the value of the fraction

Ex. 4. Compute the value of the fraction

Ans. 3.1003.

√6 √7+√3

Ans. 0.5595. √3 2√8+3√5-7√2 Ans. 0.7025.

Ex. 5. Compute the value of the fraction

9+2√10

9-2110

Ans. 5.7278.

Square Root of a Binomial Surd.

241. A binomial surd is a binomial, one or both of whose terms are surds, as 2+√3 and V5-√2.

A quadratic surd is the square root of an imperfect square. If we square the binomial surd 2+ √3, we shall obtain 7+4√3. Hence the square root of 7+4√3 is 2+√3; that is, a binomial surd of the form a±√b may sometimes be a perfect

square.

242. The method of extracting the square root of an expression of the form a±√b is founded upon the following principles:

1st. The sum or difference of two quadratic surds can not be equal to a rational quantity.

Let Va and b denote two surd quantities, and, if possible,

where c denotes a rational quantity. By transposing √ and squaring both members, we obtain

The second member of the equation contains only rational quantities, while vb was supposed to be irrational; that is, we have an irrational quantity equal to a rational one, which is impossible. Hence the sum or difference of two quadratic surds can not be equal to a rational quantity.

243. 2d. In any equation which involves both rational quantities and quadratic surds, the rational parts in the two members are equal, and also the irrational parts.

Suppose we have

x+√y=a+√b.

Then, if x be not equal to a, suppose it to be equal to a+m; then

so that

a+m+√y=a+√6,
m=√b−√y;

that is, a rational quantity is equal to the difference of two quadratic surds, which, by the last article, is impossible. Thereforex=a, and consequently √y=√b.

244. To find an expression for the square root of a±√b.

It is obvious from these equations that x and y will be rational when a2-b is a perfect square. If a2-b be not a perfect square, the values of √x and √y will be complex surds.

Hence, to obtain the square root of a binomial surd, we proceed as follows:

Let a represent the rational part, and √ the radical part, and find the values of x and y in equations (9) and (10). Then, if the binomial is of the form a+ √b, its square root will be Vx+√y. If the binomial is of the form a-√b, its square root will be √x-√y.

Verification. The square of √3+1 is 3+2√3+1=4+2√3. 2. Required the square root of 11+6√2.

Here

a=11 and √b=6√2; or b=72.
x=9 and y=2.
√x+√y=3+√2, Ans.

3. Required the square root of 11–2√30.
4. Required the square root of 2+√3.
5. Required the square root of 7+2√10.
6. Required the square root of 18+8√5.

Ans. √6–√5.

Ans. √+√.

Ans. √5+√2.

Ans. √10+2√2.

245. This method is applicable even when the binomial contains imaginary quantities.

7. Required the square root of 1+4√—3.

The required square root is therefore 2+√3, Ans.

8. Required the square root of −}+{√−3.

Ans. +√3.

Ans. 1+√=1.

9. Required the square root of 2√-1 or 0+2√—1.