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64 NOTES ON ALGEBRA.
This rule reads:
The square of the sum of any two quantities is equal to
the square of thefirst quantity plus double the product of both
quantities, plus the square of the second quantity.
(a—b)X (a —b) = (a — b) 2 = a2 —2 ab -\- b2
.
This rule reads:
The square of the difference of any two quantities is equal
to the square of thefirst quantity minus twice the product of
both quantities, plus the square of the second quantity.
(a + b) X (a — b) = a2 — b2
.
This rule reads
:
The sum of any two quantities multiplied by their differ-
ence is equal to the difference of their squares.
Extracting Roots.
An even root of a positive quantity is either + or —. An
even root cannot be extracted of a negative quantity, as \Z^
may be either a or — a; but V — a2
is impossible, because
(— a) X (— a) = a2
and (+ a) X (+ a) = a2
.
An odd root may be extracted as well of a negative quantity
as a positive quantity, and the sign of the root is always the
same as the sign of the quantity before the root was extracted.
3 3
Thus : \SaJ~= a, but \/ (— af = (— a).
Powers.
When a number or a quantity is to be multiplied by itself
a given number of times, the operation is indicated by a small
number at the right-hand corner of the quantity ; for instance,
a2 = a X a.
A quantity of this kind is called a power; the small number
is called the exponent, or the index of the power. Two powers
of the same kind may be multiplied by adding the exponents
;
for instance, a2
a3 = dMZ = ah
.
Two powers of the same kind may be divided by subtract-
ing their exponents ; for instance,
_fL = ah~2 = a3 = a X a X a.
a2
a5
—* = a5-3 = a2 = a X a.
a°~l.
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