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Notes for Week 7 Name: ______________________________ MAT 0024C 5.4 Exponent Rules and Multiplying Monomials Multiplying Monomials *Product rule for exponents: To multiply exponential expressions with the same
base, keep the common base and add the exponents. For any number x^ and any
natural numbers m^ and n^ ,
x m^ x n xm ^ n. *Multiplying monomials: To multiply two monomials, multiply the numerical factors (the coefficients) and then multiply the variable factors. Examples: Q: Simplify: 3 234. A:
Q: Multiply: ^ ^ ^
7 x^3 y^6 4 xy^32 xy^2 . A:
Q: Multiply
10 7 4.7 10 8.4 (^10) and write your answer in simplest form. A: Q: Write an expression in simplest form for the area. A: h 4h
Raising an Exponential Expression to a Power Power rule for exponents: To raise an exponential expression to a power, keep
the base and multiply the exponents. For any number x^ and any natural numbers
m and n ,
mn m n x x. Examples: Q: Simplify: 4 3 x. A: Q: Simplify: 5 47 x x. A: Powers of Products *Powers of a Product: To raise a product to a power, raise each factor of the
product to that power. For any numbers m^ and n^ , and any natural number n^ ,
xy n^ xnyn
Examples:
Q: Simplify: 4 x 2.
A:
Q: Simplify: 2 53 3 x y. A:
Multiplying Binomials *Multiplying binomials: To multiply two binomials, multiply each term of one binomial by each term of the other binomial and then combine like terms. *The FOIL method F irst O utside I nside L ast Examples:
Q: Multiply: ^2 x^ ^7 ^3 x ^8 .
A:
Q: Multiply: ^9 x^ ^3 ^ x ^9 .
A:
Q: Multiply: ^8 3 ^810 x^2 x^2 . A:
Q: A larger rectangle is formed out of smaller rectangles. a. Write an expression in simplest form for the length (along the top). b. Write an expression in simplest form for the width (along the side). c. Write an expression that is the product of the length and width that you found in parts a and b. d. Write an expression in simplest form that is the sum of the areas of each of the smaller rectangles. e. Explain why the expressions in parts c and d are equivalent. A: 5x x
Special Product: Squaring a Binomial Squaring a Binomial: The square of a binomial is the square of its first term, plus twice the product of both of its terms, plus the square of its second term.
x y 2 x^2 2 xy y^2 x y 2 x^2 2 xy y^2
Examples:
Q: Find the square: x 3 2.
A:
Q: Find the square: ^ ^
7 x 82 . A: Special Product: The Product of a Sum and Difference *Multiplying the Sum and Difference of Two Terms: The product of the sum and
difference of the two terms x^ and y^ is the square of x^ minus the square of y^.
x y x y x^2 y^2
Examples:
Q: Multiply: ^ x^ ^3 ^ ^ x ^3 .
A:
Q: Multiply: 3 1 8 3 1 8 x x. A:
5.6 Exponent Rules and Dividing Polynomials Dividing a Monomial by a Monomial *Quotient rule for exponents: To divide exponential expressions with the same base, keep the common base and subtract the exponents. For any nonzero number
x and any natural numbers m and n , where m n ,
m n n m x x x (^) . Examples: Q: Divide: 9 5 x x
A:
Q: Simplify: 12 9 y y
A:
Q: Divide: xy x y 8 24 3 2 . A: Q: Divide: (^38) 5 8 4 26 x y x y . A: Q: Divide 6 9
and write the answer in scientific notation. A:
Dividing a Polynomial by a Polynomial Examples: Q: Divide: 3 (^212) x x x . A: Q: Divide: 3 (^29) x x . A: Q: Divide: 2 2 3 4 2 2 3 x x x x . A:
