A Cubic Surd, or a Surd of the Third Order, is one with index 3; as (a+b),

7.

A Biquadratic Surd, or a Surd of the Fourth Order, is one with index 4; ás (ab), 5.

A Simple Monomial Surd Number is a single surd number, or a rational multiple of a single surd number; as √3, 2√5.

A Simple Binomial Surd Number is the sum of two simple surd numbers, or of a rational number and a simple surd number; as 2 + 3⁄4/3, 3 + √6.

5. The principles enunciated in Ch. XVI., and their proofs, hold also for irrational roots. Each principle will be restated as occasion for its use arises in this chapter. As in Ch. XVI., we shall limit the radicands to positive values, and the roots to principal roots.

Reduction of Surds.

6. A surd is in its simplest form when the radicand is integral, and does not contain a factor with an exponent equal to or a multiple of the index of the root; as 2, (ab), VaTM. A surd can be reduced to its simplest form by applying one or more of the following principles:

(iv.) In a root of a power (or a power of a root) the index of the root and the exponent of the power may both be multiplied or divided by one and the same number; or, stated symbolically,

E.g.,

Va2 = Va1; Vao = √a3.

The proof is left as an exercise for the student.

7. The following examples will illustrate the methods of reducing surds to their simplest forms:

Ex. 1. √(18 ab2) = √(9 a*b2) × √(2 a) = 3 a2b√√(2 a).

Ex. 2. (a+12+2) = (a"b2n) × (ab3) = ab2/(ab3).

Observe that the radicand is separated into two factors, one of which is a power with the highest exponent which is equal to or a multiple of the index of the required root. The result is then obtained by multiplying the rational root of this factor by the irrational root of the second factor.

= √(3a2)
_√a2 × √3 _ a√3
√4b2 √(4b2) √(4 b2)

26

When the required root of the denominator of a fraction cannot be expressed rationally, multiply both terms of the fraction by the expression of lowest degree which will make the denominator a power with an exponent equal to the index of the root.

By Art. 6 (iv.), a given surd can frequently be reduced to an equivalent surd of a lower order.

Ex. 6. (27 a3b®) = √/ba × V(3a)3 = b√(3a).

EXERCISES I.

Reduce each of the following surds to its simplest form :

Addition and Subtraction of Surds.

8. Similar or Like Surds are rational multiples of one and the same simple monomial surd, as √12, = 2√√/3, and 5√√/3. Like surds, or such surds as can be reduced to like surds, can be united by algebraic addition into a single like surd.

Ex. 1. √12+2√27−9√√/48=2√3+6√3−36√/3=−28√/3. Ex. 2. 8/40 +33/135 – 23/625 = 16 3/5 + 9 3/5 – 10 3/5

= 153/5.

Ex. 3. √2−√} +√.02 = √2 − } √2 + √ √2 = } √2.
Ex. 4. √(a3b) +2√(a3b3)+√(ab3)

= a2 √√(ab)+2 ab√(ab)+b2√(ab)=(a+b)2√(ab).

14. 2√3 −√12 + √9.

15.

24+39 – 5/192.

16. √(4a3)+√(9 a3) + √(25 a3) — √(81 a3).

17. (12 a2b)+ √(75 a2b) −√(27 a2b).

18. (64 a8b5)+ V(125 a8b5) – V(a*b5).

19. a√(a3b) + b2 √(a3b3) — 2 ab2 √(a3b3)+√(a21b25).

20. 3z (250x+z2) — 5 x (128 xz3) + 3 xz √(16 xz2).

21. 3 ab (32 a2b) + 5√(108 ab1) — ab3/(500 ab).

22. √(9 a2b2)+√(27 a3b)+5ŵ√(729 a®b2).

23. 2(3x2y) (9x+y2)+(125 x1y) — √(xy2).

24. √(9 a +27) + 3 √(4 a + 12).

25. √(4a3 +4 a2b) + √(4 ab2 + 4 63).

26. 7 x√(25a +75) — 5 √(9 x2a + 27 x2).

28. (a3b5 +3 bo) + a √ (32 a3 + 96 b) — (a3 + 3 a3b).
29. Va3-a-b-Vab2-b3-√(a + b) (a2 − b2).

Reduction of Surds of Different Orders to Equivalent Surds of the Same Order.

9. Surds of different orders can be reduced to equivalent surds of the same order by the principle given in Art. 6 (iv.):

Ex. Reduce √3, √(2 a), and √(5b) to equivalent surds of the same order.

We have

√3 = 12/36 = 12/729;

√(2 a) = 1/(2 a) 3 = 1/ (8 a3) ;

(5b) = /(5b)2 = 12/(25 b3).

Observe that the L. C. M. of the given indices is taken as the common index of the equivalent surds, and that each radicand is raised to a power whose exponent is equal to the quotient of this L. C. M. divided by the index of the given root.

10. Any rational number can be expressed in the form of a surd by writing under the radical sign a power of the number whose exponent is equal to the index.

E.g.,

2=√4 = 3/8

11. Two surds, or a surd and a rational number, can be compared by first reducing them to equivalent surds of the same order, and then comparing the resulting radicands.

Ex. Which is greater,

We have √28, and
Since 98, therefore

2 or

3?

3/3 = √/9.

98, or 3/3> √2.

EXERCISES III.

Reduce to equivalent surds of the same order:

1. √2, √5.

4. VI, VI.

7. $2, $3.

2. √3, 6.

3. √7, 10.

5. 5, 10.

6. 6, 3/4.

8. 10/15, 15/10.

10. Wa3, w/b5.

11. "+\/(x3y), " ̄\/(xy3).

13. 2, 3, 5.

14. a2, b3, V/C.

15. 2m/a3, 4m/b3, 6m/c5.

17. 5 or 10?

18.

3/25 or √11?

19. Va2 or va, when a <1?

20. √x3 or /x1, when x>1?

Which is the greatest,

21. √3, 5, or 10?

22. V, VI, or VI?

Multiplication of Surds.

12. Multiplication of Monomial Surds. The product of two or more monomial surds is found by applying the principle Va× Vb = √(ab).

Ex. 1. 53/4 × 23/6 10/24 20 3/3.

= =

If the surds are of different orders, they should first be reduced to equivalent surds of the same order.

Ex. 2. √√/a × √/a2 = √/a3 × √/a* = &/a2 = a&/a.

Ex. 3.

† (a2b) × √(a3b2) × √(a*b3) = W/ (ab1) × 12/ (a°b®) × 12/(ab1o)

=12/(a2562) = a2b12/(ab3).