9. 25 x1- 44x3 − 40 x + 4x + 25 + 46 x2 – 12 x3.

10. 446.0544.

11. 811440.64.

12. .68112009.

Find the cube roots of the following:

·54ab9ab2 + 28 a3b3 — 3a2b1 — 6 ab3 — bo.

15. 10x+12-1-3x-6x-12x+x+3x+6x-10.

16. 31855.013. 17. 1039.509197. 18. .000152273304.

19. Find the fourth root of

2-8x+16x+16x56x-32x2+64x2+64x+16.

20. Find the sixth root of

a12-6 a1b+15ab2-20 ab3 +15 ab1-6a2b3 + bε.

21. Find the fifth root of

2-10-10x+302-120x6+48x+ 240x240 x2-160x-32.

22. Find the fourth root of 209727.3616.

23. Find the sixth root of .009474296896. 24. Find the fifth root of 281530.56843.

XVII. SURDS; THE THEORY OF EXPONENTS.

269. A number is said to be rational when it is either a positive or negative integer or fraction, or a number which can be reduced to one of these forms.

Only rational numbers have been considered in the preceding chapters; and every letter has been understood as representing a rational number, unless the contrary has been expressly stated (Art. 33, Note).

We shall now proceed to define a surd.

270. If n is any positive integer, and a a rational number (Art. 269) which is not a perfect power of the nth degree, and which is positive if n is even, the expression Va is said to be a surd.

An expression is also said to be a surd when it can be reduced to the above form.

271. Let us consider the surd √2.

It is impossible to find a rational number whose square shall be equal to 2; but it is possible to find two rational numbers which shall differ from each other by less than any assigned number, however small, and whose squares shall be less and greater than 2, respectively.

For, writing the squares of the consecutive integers 1, 2, etc., we have 121, 224, etc.; hence 1 and 2 are two numbers which differ by unity, and whose squares are less and greater than 2, respectively.

=

Again, we have 1.121.21, 1.22 1.44, 1.32 = 1.69, 1.42

1.96, 1.52= 2.25, etc.; hence 1.4 and 1.5 are two numbers which differ by .1, and whose squares are less and greater than 2, respectively.

=

Again, 1.4121.9881, 1.422 2.0164, etc.; hence 1.41 and 1.42 are two numbers which differ by .01, and whose squares are less and greater than 2, respectively.

It is evident that, by sufficiently continuing the above process, two numbers may be found which shall differ from each other by less than any assigned number, however small, and whose squares shall be less and greater than 2, respectively.

272. In general, if n and a have the same meanings as in Art. 270, it is possible to find two rational numbers which shall differ from each other by less than any assigned number, however small, and whose nth powers shall be less and greater than a, respectively.

273. The successive numbers in the example of Art. 271 whose squares are less than 2, are 1, 1.4, 1.41, etc.; and the numbers whose squares are greater than 2, are 2, 1.5, 1.42, etc.

If each series is continued to r terms, the difference between the rth terms of the two series can be made less than any assigned number, however small, by sufficiently increasing r.

But √2 is intermediate in value between the rth terms of the two series, and hence the difference between the rth term of either series and √2 can be made less than any assigned number, however small, by sufficiently increasing r Therefore the rth term of either series approaches √2 as a limit (Art. 209) when r is indefinitely increased.

274. In general, if n and a have the same meanings as in Art. 270, and a1', ag', as', etc., is a series of rational numbers whose nth powers are less than a, and a", a", a", etc., a series of rational numbers whose nth powers are greater than a, such that a1"~a,'=1, a‚"~aq'= .1, a3′′~ az'= .01, etc., it may be shown, as in Art. 273, that the rth term of either series approaches Va as a limit when r is indefinitely increased.

275. The definitions of Addition and Multiplication, as given in Arts. 34, 38, 50, and 53, hold only when the numbers involved are rational; it then becomes necessary to give definitions for Addition and Multiplication when any or all of the numbers involved are surds.

Let n and p be positive integers; let a be a rational number which is not a perfect power of the nth degree, and which is positive if n is even; and let b be a rational number which is not a perfect power of the pth degree, and which is positive if p is even.

Let a', a', be a series of rational numbers whose nth powers are less than a, and a", a", ..., ɑ,', ..., a series of rational numbers whose nth powers are greater than a, such that a," a = 1, a" ~ a' .1, ..., a,"~a,' =(.1)',....

....

Also, let b', b2', b,' be a series of rational numbers whose pth powers are less than b, and b1", b2'', ..., b,'', ..., a series of rational numbers whose pth powers are greater than b, such that b," b' 1, b."~ b'= .1, = (.1)'-', ... .

b," ~ b,'

is

Then, to add a and b is to find the limit, when indefinitely increased, of a,' + b,'; and to multiply Va by b is to find the limit, when r is indefinitely increased, of a,'x b'.

A meaning similar to the above will be attached to any expression which is not a rational number, and which is the result of any finite number of the following operations performed upon one or more rational numbers or surds:

1. Addition or Subtraction. 2. Multiplication or Division. 3. Raising to any power whose exponent is a positive integer.

276. We gave in Chapter II. proofs of the fundamental laws of Algebra in all cases where the numbers involved were rational; we will now show how to prove these laws when any or all of the letters involved represent surds.

Let it be required, for example, to prove the Commutative Law for Multiplication (Art. 58) with respect to the product of two surds, a and b, where n, p, a, and b have the same meanings as in Art. 275.

That is, to prove

Vax Vb = Vox Va.

With the notation of Art. 275, Vax Vb is the limit, when is indefinitely increased, of a'x b,', and b× Va is the limit, when r is indefinitely increased, of b,'x a'. But by Art. 58, since a,' and b,' are rational numbers,

and since a, b,' and b,' × a,' are functions of r which are equal for every positive integral value of r, by Art. 213 their limits when r is indefinitely increased are equal.

Note. It must be borne in mind, as stated in Art. 122, Note, that the above equation means simply that the product of one of the nth roots of a and one of the pth roots of b is equal to the product of one of the pth roots of b and one of the nth roots of a; and a similar interpretation must be given to every result involving surds.

In like manner, any result in Chapter II. may be proved. to hold when any or all of the letters involved represent surds.

For example, the equations (Arts. 42, 67),

hold for any rational or surd values of a and b; which shows that the definitions of the operations of subtraction and division, as given in Arts. 41 and 67, hold for any rational or surd values of the numbers involved.

It follows from the above that every statement or rule, in Chapters III. to XVI. inclusive, in regard to expressions where any letter involved represents any rational number whatever, holds equally when this letter represents a surd.