Algebraical Formulæ

for the Use of Candidates for Responsions

Meaning of indices $a^{x}$	$= a \times a \times a \times &c.$ (`x` factors).
”” $a^{1}$	$= a$ .
Multiplication $a^{x} \times a^{y}$	$= a^{x + y}$ .
Division $a^{x} \div a^{y}$	$= a^{x - y}$ .
hence $a^{0}$	$= 1$ .
$a^{- x}$	$= \frac{1}{a^{x}}$ .
$a^{- 1}$	$= \frac{1}{a}$ .
Involution (single term) ${(a^{x})}^{y}$	$= a^{x y}$ .
Evolution ” $\sqrt[y]{a^{x}}$	$= a^{x / y}$ .
”” $\sqrt[y]{a}$	$= a^{1 / y}$ .
Involution (two terms)
(1) ${(a + b)}^{2}$	$= a^{2} + 2 a b + b^{2}$ .
(2) ${(a - b)}^{2}$	$= a^{2} - 2 a b + b^{2}$ .
(3) ${(a + b)}^{3}$	$= a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}$ .
(4) ${(a - b)}^{3}$	$= a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}$ .
(5) ${(a + b)}^{n}$	$= a^{n} + n a^{n - 1} b + &c.$ , where the index of `a` continually decreases by unity, while that of `b` increases, and where the coefficient of each term is formed from the preceding by the rule “Multiply together the coefficient and the index of `a` and divide by the place of the term.”
(6) ${(a - b)}^{n}$	$= a^{n} - n a^{n - 1} b + &c.$ , where each term is formed as in the last case, and the signs are alternately + and −.
Resolution into factors
(1) $a^{2} + b^{2}$	has no factors.
(2) $a^{2} - b^{2}$	$= (a + b) . (a - b)$ .
(3) $a^{3} + b^{3}$	$= (a + b) . (a^{2} - a b + b^{2})$ .
(3) $a^{5} + b^{5}$	$= (a + b) . (a^{4} - a^{3} b + a^{2} b^{2} - a b^{3} + b^{4})$ .
and generally, if `n` be an odd prime,
(3) $a^{n} + b^{n}$	$= (a + b) . (a^{n - 1} - a^{n - 2} b + \dots)$ , where the index of `a` continually decreases by unity, while that of `b` increases, and the signs are alternately + and −.
(4) $a^{3} - b^{3}$	$= (a - b) . (a^{2} + a b + b^{2})$ .
(4) $a^{5} - b^{5}$	$= (a - b) . (a^{4} + a^{3} b + a^{2} b^{2} + a b^{3} + b^{4})$ .
and generally, if `n` be an odd prime,
(4) $a^{n} - b^{n}$	$= (a - b) . (a^{n - 1} + a^{n - 2} b + \dots)$ , where the indices are as in the former case, and the signs are all +.
If `T` be a power of 2,
(5) $a^{T} + b^{T}$	cannot be resolved.
(6) $a^{T} - b^{T}$	can be resolved by form (2).
If `N` contain odd prime factors only,
(7) $a^{N} + b^{N}$	can be resolved by form (3).
(8) $a^{N} - b^{N}$	””(4).
If `M` be even and contain an odd prime factor,
(9) $a^{M} + b^{M}$	can be resolved by form (3).
(10) $a^{M} - b^{M}$	””(2) or (4).
If `x`, `y` be commensurable,
(11) $a^{x} + b^{y}$ }	can be put into one of the first four forms,
(12) $a^{x} - b^{y}$	according to the nature of the G. C. M.