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NOTES ON ALGEBRA. 65
Thus, any quantity in power must be 1, because always
when dividend and divisor are alike the quotient must be 1.
—— = #5-6 = a-1
, but ab
divided by a6
is equal to 1 ; therefore
SL must be J_ ;
lL.= <r+ = -A-
a6
a a’ a2
Thus, any quantity with a negative exponent is one divided
by that quantity considering the exponent as positive. We
may, therefore, say that as a positive exponent indicates how
many times a quantity is to be used as a factor, a negative ex-
ponent indicates how many times a quantity should be used as
a divisor ; for instance,
a i
= ; a l = ; a 3 =
a a X a a Xa X a
Thus
:
6-i = |; 6-2 = (i-) 2 = A ; S-2 = (i)2 = 1 etc.
Equations.
An algebraic expression of equality between two quantities
is called an equation. The two quantities are connected by the
sign of equality, and the quantity on the left-hand side of the
sign is called the first member, and the quantity on the right-
hand side is called the second member of the equation.
If a part or all of the known quantities are expressed in
letters it is called a literal equation: ax = a + b — 2<: is a
literal equation.
If the equation contains no letters except the unknown
quantity, usually expressed by x, it is called a numerical equa-
tion ; for instance, 5 x = 12 — 7 + 9> is a numerical equation.
In solving equations, we may, without destroying the equal-
ity of the equation, add an equal quantity to both members,
subtract an equal quantity from both members, multiply both
members by an equal quantity, divide both members by an
equal quantity, extract the same root of both members, raise
both members to the same power.
Quantities inclosed by parentheses, a bar, or under a radical
sign, and quantities connected by the sign of multiplication,
must always be considered and operated upon as one quantity,
Example 1.
^ = 3X8 + 10-3 + 3X10.
x = 24 + 10 — 3 + 30.
x = 64— 3.
* = 61. .
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