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

root required, or x, increased by + ; and we shall shew 3a2

a

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, a 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.

7. xo - 6αx3 +15a2x2 - 20a3x3 + 15α*x2 - 6a3x + a3.

9. 4 {(a2 — b3) cd + ab (c3 — d2)}3 + {(a3 — b3) (c3 — d3) — 4abcd}3. 10. a*+b2 + c* + d1 — 2a2 (b2 + d3) — 2b3 (c2 – d3) + 2c3 (a3 — d3).

2

14. a*+2(2b-c) a3 + (4b3 — 4bc + 3c2) a2 + 2c2 (2b - c) a + c*. 15. (α – 2b)2 x1 — 2a (a — 2b) x3 + (a3 + 4ab − 6a — 862 + 12b) x2 − (4ab – 6a) x + 46a — 126 + 9.

16.

--

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

18. 8x+48cx2 + 60c2x1 – 80c3ñ3 – 90c1x2 + 108c3x − 27co.

31189187729.

970.645048.

371742108367626890260631.

2

2

tract the fourth root of (a2 + 1) - 4 (x + 1) + 12.

1 +12.

a number contain n digits, its square root contains -1)"} digits.

ew that the following expression is an exact square: - (y3 — zx)3 + (≈2 — xy)3 — 3 (x3 — yz) (y3 — zx) (z2 — xy).

XVIII. THEORY OF INDICES.

e have defined a", where m is a positive integer, as f m factors each equal to a, and we have shewn that

n.

α

Hitherto then an exponent has always been a r; it is however found convenient to use exponents #positive integers, and we shall now explain the ch exponents.

fractional indices and negative indices have not yet ve are at liberty to give what definitions we please it is found convenient to give such definitions to nake the important relation aTM × a" = am+n always n and n may be.

e; required the meaning of a31.

Eion we are to have a1x aaa. Thus a

must

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, a

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

These examples would enable the student to understand wha 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 of a; that is, 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

am xa ̄" = am-n.

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,