REDUCTION OF SURDS.

236. Transformation of Surds of Any Order into Surds of a Different Order having the Same Value. A surd of any order may be transformed into a surd of a different order having the same value. Such surds are said to be equivalent.

237. Surds of different orders may therefore be transformed into surds of the same order. This order may be any common multiple of each of the given orders, but it is usually most convenient to choose the least common multiple.

Ex. Express Va3, V/b2, Va5 as surds of the same lowest order.

The least common multiple of 4, 3, 6 is 12; and expressing the given surds as surds of the twelfth order they become a9, 12/b8, 12/a10.

238. Surds of different orders may be arranged according to magnitude by transforming them into surds of the same order.

Ex. Arrange √3, 3/6, †/10 according to magnitude.

The least common multiple of 2, 3, 4 is 12; and, expressing the given surds as surds of the twelfth order, we have

Hence arranged in ascending order of magnitude the surds are √3, 10, 3/6.

EXAMPLES XXIII. a.

Express as surds of the twelfth order with positive indices:

Express as surds of the nth order with positive indices :

239. Reduction of a Surd to its Simplest Form. The root of any expression is equal to the product of the roots of the separate factors of the expression.

Hence it appears that a surd may sometimes be expressed as the product of a rational quantity and a surd; when the surd factor is integral and as small as possible, the surd is in its simplest form [Art. 233].

Thus the simplest form of √128 is 8√2.

Conversely, the coefficient of a surd may be brought under the radical sign by first raising it to the power whose root the surd expresses, and then placing the product of this power and the surd factor under the radical sign.

EXAMPLES.

(1) 7√5 = √49 · √5 = √245.

(2) a/b/a/bab.

In this form a surd is said to be an entire surd [Art. 232].

By the same method any rational quantity may be expressed in the form of a surd. Thus 2 may be written as √4, and 3 as

27.

240. When the surd has the form of a fraction, we multiply both numerator and denominator by such a quantity as will make the denominator a perfect power of the same degree as the surd, and then take out the rational factor as a coefficient.

EXAMPLES. (1) √} = √} = √2 × } = {√2.

Express (1) as entire surds, (2) in simplest form :

ADDITION AND SUBTRACTION OF SURDS.

241. To add and subtract like surds: Reduce them to their simplest form, and prefix to their common irrational part the sum of the coefficients.

Ex. 2. The sum of x V8 x3a + y V — y3a — z Vz3a

242. Unlike surds cannot be collected.

Thus the sum of 5 √2, −2√3, and √6 is 5√2−2√3 +6, and cannot be further simplified.

EXAMPLES XXIII. c.

Find the value of 1. 3√45-√20 + 7 √5. 2. 4√63+57-828. 3. √44 - 5 √ 176 + 2 √99. 4. 2√3635 √243 + √192. 5. 23/189 + 3 3/875 – 7 3/56. 6. 53/8173/192 + 4 3/648.

7. 3/1627/32 + /1250. 8. 5-54-2 V-16+4/686. 9. 4√128 +4 √75 – 5 √162. 10. 5√24-2 √54 – √6. 11. √252 — √294 – 48 √ž. 12. 3√147√} −√}.

MULTIPLICATION OF SURDS.

243. To multiply two surds of the same order: Multiply separately the rational factors and the irrational factors.

(3) Va + bx Va − b = √(a + b) ( a − b) = √ a2 — b2.

If the surds are not in their simplest form, it will save labor to reduce them to this form before multiplication.

Ex. The product of 5 √32, √48, 2 √ 54

= 5 • 4 √2 × 4 √3 × 2 · 3 √6 = 480 · √2 · √3 · √6 = 480 × 6 = 2880.

244. To multiply surds of different orders: Reduce them to equivalent surds of the same order, and proceed as before.

245. Suppose it is required to find the numerical value of the quotient when √5 is divided by √7.

At first sight it would seem that we must find the square root of 5, which is 2.236..., and then the square root of 7, which is 2.645..., and finally divide 2.236... by 2.645...; three troublesome operations.

But we may avoid much of this labor by multiplying both numerator and denominator by √7, so as to make the denominator a rational quantity. Thus

246. The great utility of this artifice in calculating the numerical value of surd fractions suggests its convenience in the case of all surd fractions, even where numerical

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