Operations with Monomials
A monomial is an algebraic expression that is a constant or a
product of a constant and one or
more variables with whole number exponents.
Monomials:
Basic Rules for Exponents
1. Extending the Multiplying of â€œpowers with the same baseâ€: a^{ m}
Â· a^{ n}
Â· a^{ p}
=a^{ m + n + p}
Add exponents by the rules of adding signed numbers where m, n, p
{ integers}
Example 1:
a) z^{ 5} Â· z^{ 2}
Â· z^{ 4} = z ^{5+2+4} = z^{ 11}
b) x^{ 2} Â· x^{ 3}
Â· x^{ 4}
Â· x^{ 5} = x^{ 2+3+4+5 }= x
^{14}
2. Powering of powers:
For base a and exponents m, n
{ natural numbers
}
(a^{ m})^{ n} = a^{ mÂ·n}
Example 2:
a) (x^{ 4} )^{3} = x^{ 4Â·3} = x^{ 12}
b) (y^{ 2})^{ 5} = y^{2Â·5} = y^{ 10}
3. Powering of products with different bases a and b, and exponent n
{ natural numbers}
(aÂ·b)^{ n} = a^{
n} Â· b^{ n}
NOTE: The exponent affects only the number that it
immediately touches so you must always put
negative numbers in parentheses ( ).
Example 3:
1. (x ^{5} y^{ 7})^{3} = (x ^{5})^{3}Â·(y
^{7})^{3} = x^{5Â·3}Â·y^{7Â·3} = x^{15} y^{21} Apply properties and multiply exponents.
2. (2 Â· x^{2}y^{3})^{4} = (2)^{4}
Â·(x^{2})^{4}Â·(y^{3})^{4} = 16
Â·(x^{2Â·4})Â·(y^{3Â·4}) = 16
Â· x^{8}y^{12} Apply properties, multiply exponents.
Definition of Negative Exponent
Note: a ^{n} Ã— a ^{n} = 1 They are reciprocals.
The sign in the exponent does not affect the sign of the number in the base.
Example 4:
a)
Since 2 is in the denominator place positive in the numerator:
b) x ^{8} Since 8 is in the numerator place positive in the denominator:
Definition of Zero Exponent a^{ 0} = 1 Note: The result of raising any base to the 0 exponent
will always be 1. (a ≠ 0)
1. (57x^{9} y^{32} z ^{−25} )^{0} = 1
4. Division of powers with positive exponents:

Subtract the smaller exponent from the larger
exponent and put the resulting â€œpower â€ in the â€œplaceâ€
of the larger â€œpowerâ€.
Put a 1 in the numerator if the larger â€œpowerâ€ is in the
denominator.

Example 5: (Working with positive exponents):
1.
[7 > 3 result goes in the numerator.]
2.
[9 > 5 result goes in the denominator.]
3.
[4 = 4 Exponents the same result is always 1.]
