Powers of monomials

key notes:

Definition:

• A monomial is an algebraic expression consisting of a single term.
• The power of a monomial is determined by the exponent of the variable in that term.

General Form:

• A monomial in the form ���axn, where �a is the coefficient, �x is the variable, and �n is the exponent.

Rules for Multiplying Monomials:

1. Product of Coefficients: �⋅�=��a⋅b=ab
2. Product of Variables: ��⋅��=��+�xn⋅xm=xn+m
3. Product of Powers with the Same Base: (��)�=���(xn)m=xnm

Rules for Dividing Monomials:

1. Quotient of Coefficients: ��=��ba​=ba​
2. Quotient of Variables: ����=��−�xmxn​=xn−m
3. Quotient of Powers with the Same Base: ����=��−�xmxn​=xn−m

Power of a Monomial Raised to a Power:

• (���)�=��⋅���(axn)m=am⋅xnm

Zero Exponent:

• �0=1x0=1 for any nonzero value of �x.

Negative Exponent:

• �−�=1��x−n=xn1​

Examples:

1. 2�3⋅3�2=6�52x3⋅3x2=6x5
2. 4�42�2=2�22x24x4​=2x2
3. (2�3)2=4�6(2x3)2=4x6
4. 5�3⋅5�−2=25�5x3⋅5x−2=25x

Practice Tips:

• Be careful with signs when multiplying or dividing monomials.
• Understand the role of exponents in combining monomials.
• Practice simplifying expressions involving the powers of monomials.

Learn with an example

let’s practice!🖊️