Monomials Multiplying Monomials and Raising Monomials to Powers

Vocabulary Monomials - a number, a variable, or a product of a number and one or more variables 4x, 20x 2 yw 3, -3, a 2 b 3, and 3yz are all monomials. Constant – a monomial that is a number without a variable. Base – In an expression of the form x n, the base is x. Exponent – In an expression of the form x n, the exponent is n.

Writing - Using Exponents Rewrite the following expressions using exponents: The variables, x and y, represent the bases. The number of times each base is multiplied by itself will be the value of the exponent.

Writing Expressions without Exponents Write out each expression without exponents (as multiplication): or

Simplify the following expression: (5a 2 )(a 5 )  Step 1: Write out the expressions in expanded form.  Step 2: Rewrite using exponents. Product of Powers There are two monomials. Underline them. What operation is between the two monomials? Multiplication!

For any number a, and all integers m and n, a m a n = a m+n. Product of Powers Rule

If the monomials have coefficients, multiply those, but still add the powers. Multiplying Monomials

These monomials have a mixture of different variables. Only add powers of like variables. Multiplying Monomials

Simplify the following: ( x 3 ) 4 Note: 3 x 4 = 12. Power of Powers The monomial is the term inside the parentheses.  Step 1: Write out the expression in expanded form.  Step 2: Simplify, writing as a power.

Power of Powers Rule For any number, a, and all integers m and n,

Monomials to Powers If the monomial inside the parentheses has a coefficient, raise the coefficient to the power, but still multiply the variable powers.

Monomials to Powers (Power of a Product) If the monomial inside the parentheses has more than one variable, raise each variable to the outside power using the power of a power rule. (ab) m = a m b m

Monomials to Powers (Power of a Product) Simplify each expression:

What about dividing monomials?

1.Be able to divide polynomials 2.Be able to simplify expressions involving powers of monomials by applying the division properties of powers.

Simplify: Step 1: Rewrite the expression in expanded form Step 2: Simplify. For all real numbers a, and integers m and n: Remember: A number divided by itself is 1.

Simplify: Step 1: Write the exponent in expanded form. Step 2: Multiply and simplify. For all real numbers a and b, and integer m:

Apply quotient of powers. Apply power of a quotient. Apply quotient of powers Apply power of a quotient Simplify Apply power of a power

1. 2. THINK! x 3-3 = x 0 = 1