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# 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 = y2Â·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)3Â·(y 7)3 = x5Â·3Â·y7Â·3 = x15 y21 Apply properties and multiply exponents.

2. (-2 Â· x2y3)4 = (-2)4 Â·(x2)4Â·(y3)4 = 16 Â·(x2Â·4)Â·(y3Â·4) = 16 Â· x8y12 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. (57x9 y32 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.]