PROBLEM IX.

To find the roots of surd quantities.

(171.) A root of a quantity is a factor which, multiplied by itself a certain number of times, will produce the given quantity. But we have seen that a radical quantity is involved by multiplying its exponent by the exponent of the required power. Hence,

To find the roots of surd quantities,

Divide the fractional exponent by the index of the required

root.

Thus, the square root of a3 is a12=a".

For, by Art. 169, we obtain the square of a by multiplying 11

the exponent by 2;

To find multipliers which shall cause surds to become rational. (172.) I. When the surd is a monomial.

The quantity va is rendered rational by multiplying it by

va.

For

√ax√a=a3×a2=a.

y a3.

So, also, a is rendered rational by multiplying it by

In general, a" is rendered rational by multiplying it by a

Multiply the surd by the same quantity having such an ex ponent as, when added to the exponent of the given surd, shall be equal to unity.

(173.) II. When the surd is a binomial.

If the binomial contains only the square root, multiply the given binomial by the same expression with the sign of one of its terms changed, and it will give a rational product.

Ex. 1. The expression va+vb

Ex. 2. Find a multiplier which shall render 5+ √3 rational

These two examples are comprehended under the Rule in Art. 62, the product of the sum and difference of two quantities is equal to the difference of their squares.

Ex. 3. Find a multiplier that shall make 5+ √3 rational, and determine the product.

Ex. 4. Find a multiplier that shall make ✓5-√x rational and determine the product.

Ex. 5. Find a multiplier that shall make va- √ abc r tional.

III. When the surd is a trinomial, it may be reduced, by successive multiplications, first to a binomial surd, and then to a rational quantity.

Ex. 1. Find multipliers that shall make 5+√3-√2 rational.

Second product,

60+12/15

-12/15-36

60-36-24, a rational quantity.

Ex. 2. Find multipliers that shall make va+b+c ra tional, and determine the product.

PROBLEM XI. ·

(174.) To reduce a fraction containing surds to another having a rational numerator or denominator.

RULE.

Multiply both numerator and denominator by a factor which will render either of them rational, as the case may require.

Ex. 1. If both terms of the fraction be multiplied by

α

va
vb

✔a, it will become ab' in which the numerator is rational.

√ ab b

Or if both terms be multiplied by b, it will become in

which the denominator is rational.

Ex. 7. Reduce 5+2 to a fraction having a rational numerator.

(175.) The utility of the preceding transformations may be illustrated by computing the numerical value of a fractional surd.

Ex. 1. Suppose it is required to find the square root of &;

that is, it is required to find the value of the fraction

If we make the denominator rational, we shall have

which it is only necessary to extract the square root of the numerator, and the value of the fraction is found to be 0.6546.

Ex. 2. It is required to find the value of the fraction

Making the denominator rational, we have value of which is 3.1003.

Ex. 3. Required the value of the expression

8

Ex. 4. Required the value of the expression

√3
2√8+3√5-7√2°

Ex. 5. Required the value of the expression

Ans. 0.7025.

9+2/10 9-2/10 Ans. 5.7278.

PROBLEM XII.

(176.) To free an equation from radical quantities.

This may generally be done by successive involutions. For this purpose, we first free the equation from fractions. If there is but one radical expression, we bring that to stand alone on one side of the equation, and involve the whole equation to a power denoted by the index of the radical.

Clearing of fractions, and transposing a, we obtain

√2ax+x=ab-a.

The square of this equation is

2ax+x2=a2b2—2a3b+a2,

which is free from radical quantities.

Ex. 2. Free the equation

2a2

x+√a2+x2=

√a2+x2

from radical quantities.

If the equation contains two radical expressions, combined with other terms which are rational, it will generally be best to bring one of the radicals to stand alone on one side of the equation before involution. One of the radicals will thus be made to disappear, and, by repeating the operation, the remaining radical may be exterminated.