Exact Division, Negative Exponents
Algebra Help: Section 2.24
Learn about exact division, reciprocals, zero powers and negative exponents, with algebra.
Exact Division
An Exact Division is one in which the quotient has no fractional part.
From the rule for division it's evident that the exact division of one mononomial by another will be impossible.
Abbreviated rule for division from section 5.2.
Division, considered in its most elementary sense, is not merely the converse of multiplication; it is a short process of finding how many times one quantity can be subtracted from another of the same kind. When the subtraction is possible, and diminishes the numeral value of the minuend, and brings it nearer to zero, the operation is real, and must be marked plus. When the subtraction is not possible without going farther from zero, we take the converse operation, marked minus.
Exact division of one mononomial by another's impossible when:
- The coefficient of the divisor is not exactly contained within the coefficient of the dividend.
- A literal factor has a greater exponent in the divisor than in the dividend.
- A literal factor of the divisor is not found in the dividend.
It's also evident that the division of one polynomial by another will be impossible when:
- The first term of the divisor, arranged with reference to any one of its letters, is not exactly contained in the first term of the dividend, arranged with reference to the same letter.
- A remainder occurs, having no term which will exactly contain the first term of the divisor.
In all these cases exact division is impossible, the quotient may be indicated by writing the dividend above a horizontal line, and the divisor below.
Reciprocals, Zero Powers & Negative Exponents
The Reciprocal of a quantity is 1 divided by that quantity;
thus 1/a is the reciprocal of a.
Also 1/(x - y) is the reciprocal of x - y.
If, in the division of powers, we conform strictly to the rule of subtracting the exponent of the divisor from the exponent of the dividend, then, in the case of equal powers, the exponent of the quotient will be 0. In the cases where the divisor is the higher power, the exponent of the quotient will become negative.
To explain the import of a zero when used as an exponent, we observe that the quotient of any quantity divided by itself is 1. Consequently, when the divisor and dividend are like powers of the same quantity, we may have two expressions for the quotient; thus
a a - = a 1 - 1 = a0, or -- = 1 a a am am - = a m - m = a0, or -- = 1 am am
Therefore, a0 = 1.
NOTE - In the cases previously stated how many the quotient be written? Define the reciprocal of a quantity. In division when will the exponent of the quotient be 0? When negative? Explain why. What is the value of any quantity whose exponent is 0? Why?
Literal a may represent any quantity whatever.
Therefore:
Any quantity having a zero for an exponent is equal to unity; or one.
NOTE - When a quantity with a zero for an exponent is a factor in an algebraic expression, it may be suppressed without affecting the value of the expression; yet it is frequently retained in order to indicate the process by which the result was obtained.
To show the signification of negative exponents, let us
divided a5 by a7.
Take the difference of the exponents.
Therefore
a5 ÷ a7 = a 5-7 = a-2.
The value of the quotient will not be altered if we divide
both dividend and divisor by a5. Therefore:
1
a5 ÷ a7 = 1 ÷ a2 = -
a2
a5 1 -- = -- a7 a2
These quotients are equal:
1
a-2 = -
a2
This principle may also be illustrated as follows.
a0 -- = a0-2 = a-2 a2
a0 1 -- = - a2 a2
Therefore:
1
a-2 = -
a2
The same principle with a literal exponent, follows.
a0 -- = a0-m = a-m am
a0 1 -- = - am am
Therefore:
1
a-m = -
am
From these illustrations we deduce the following inference:
Any quantity having a negative exponent is equal to the reciprocal of that quantity with an equal positive exponent.
NOTE - What do negative exponents signify? What relation do they bear to reciprocals?
Page 64-66
Page 64-66
Help with Algebra Homework
Free tutorials provide math help for math problems. Tutorials start with algebra 1. Algebra tutorials include word problems related to algebra topics, as they're introduced. See the menu, above for algebra math equations, and explanation, by section.
Windows algebra games, designed to help with classwork, coming soon!
Windows games, in progress, work with this set of free tutorials, modified
from the public domain text book titled,
New Elementary Algebra
containing the rudiments of the science for schools and academies.
By Horation N Robinison, LL. D. Ivison, Blakeman & Company, Publishers, New York and Chicago.
