257. When n+2 figures of a cube root have been obtained by the ordinary method, n more may be obtained by division only, supposing 2n + 2 to be the whole number.

Let N represent the number whose cube root is required, a the part of the root already obtained, x the part which remains to be found; then

Thus N – a3 divided by 3a2 will give the rest of the cube

x2 root required, or x, increased by + ; and we shall shew a 3a

that the latter expression is a proper fraction, so that by neglecting the remainder arising from the division, we obtain the part required. For by supposition, x is less than 10", and a is not

EXAMPLES OF EVOLUTION.

Extract the square roots of the expressions contained in the following examples from 1 to 15 inclusive.

9. 4 {(a2 − b3) cd + ab (c2 — d2)}2 + {(a2 — b3) (c3 — d3) — 4abcd}3.

10. a*+b*+ c*+ d* — 2a2 (b2 + d2) — 262 (c2 — d2) + 2c3 (a2 – đ3).

14.

4

+

х

ax 2+

a1+2 (2b - c) a3 + (4b3 — 4bc + 3c2) a2 + 2c2 (2b – c) a + c3. 15. (a -26)2 x* − 2a (a − 2b) x3 + (a2 + 4ab — 6a – 862 + 126) x2

16.

4

− (4ab — 6a) x + 46o – 126 + 9.

Find the square root of the sum of the squares of ·2, ·4,

•6, 86.

Extract the cube root of the expressions and numbers in the following examples from 17 to 24 inclusive.

17. 8x6-36x + 66x1 – 63x3 + 33x2 - 9x+1.

18. 8x+48cx2 + 60c2x2 - 80c3x3- 90c⭑x2 + 108cx-27c..

8x - 36cx5+ 102c2x2 – 171c3×3 + 204c1x2 – 144c3x + 64co.

167.284151.

19.

20.

21. 731189187729.

25. If a number contain n digits, its square root contains {2n + 1 − (− 1)"} digits.

-

26. Shew that the following expression is an exact square :

(x2 — yz)3 + (y2 — zx)3 + (≈2 —. xy)3 — 3 (x2 — yz) (y2 — zx) (≈2 — xy).

XVIII. THEORY OF INDICES.

258. We have defined a", where m is a positive integer, as the product of m factors each equal to a, and we have shewn that

or less than n. Hitherto then an exponent has always been a positive integer; it is however found convenient to use exponents which are not positive integers, and we shall now explain the meaning of such exponents.

259. As fractional indices and negative indices have not yet been defined, we are at liberty to give what definitions we please to them; and it is found convenient to give such definitions to them as will make the important relation a” × a" = aTM+n always true, whatever m and n may be.

For example; required the meaning of aa.

Thus a must

By supposition we are to have a xaa = a. be such a number that if it be multiplied by itself the result is a ;

and the square root of a is by definition such a number; therefore a must be equivalent to the square root of a, that is, a3 = a* = √a.

Again; required the meaning of a3.

By supposition we are to have

a3× a3× a3 = a3+§+§ — a2 = a.

Hence, as before, a3 must be equivalent to the cube root of a,

These examples would enable the student to understand what is meant by any fractional exponent; but we will give the definition in general symbols in the next two Articles.

260. Required the meaning of a where n is any positive whole number.

therefore a must be equivalent to the nth root of a,

261. Required the meaning of a" where m and n are any positive whole numbers.

therefore a" must be equivalent to the nth root of a”,

Hence a' means the nth root of the mth power

in a fractional index the numerator denotes a power and the denominator a root.

262. We have thus assigned a meaning to any positive index, whether whole or fractional; it remains to assign a meaning to negative indices.

For example, required the meaning of a ̄3.

We will now give the definition in general symbols.

263. Required the meaning of a""; where n is any positive number whole or fractional.

By supposition, whatever m may be, we are to have

Now we may suppose m positive and greater than n, and then, by what has gone before, we have

In order to express this in words we will define the word reciprocal. One quantity is said to be the reciprocal of another when the product of the two is equal to unity; thus, for example, x is the reciprocal of 1.