MultDivideMonomials.notebook
1
September 18, 2014
Powers and Exponents
Lesson objectives Teachers' notes
1) Power Rules
2) Multiplying and Dividing polynomials with monomialsMultiplying binomials
3) Working with Negative Exponents
4) Multiplying Power to a power
MultDivideMonomials.notebook
2
September 18, 2014
Teachers' notesLesson objectives
Subject:
Topic:
Grade(s):
Prior knowledge:
Mathematics
Powers and Exponents
Grade 8
Integers and Variables
Lesson notes:
8.EE.1: Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3 2 × 3–5 = 3–3 = 1/33 = 1/27.
MultDivideMonomials.notebook
3
September 18, 2014
Multiplying and Dividing MonomialsBut what is a monomial?
Monomials: a number, a variable, or product of a number and one or more variable(s) with nonnegative integer exponents.
MultDivideMonomials.notebook
4
September 18, 2014
Monomials: a number, a variable, or product of a number and one or more variable(s) with nonnegative integer exponents.
Example: h2, x, 10
Multiplying and Dividing MonomialsBut what is a monomial?
MultDivideMonomials.notebook
5
September 18, 2014
Monomials: a number, a variable, or product of a number and one or more variable(s) with nonnegative integer exponents.
Example: h2, x, 10
Constant: a monomial that is a real number.
Multiplying and Dividing MonomialsBut what is a monomial?
MultDivideMonomials.notebook
6
September 18, 2014
Monomials: a number, a variable, or product of a number and one or more variable(s) with nonnegative integer exponents.
Example: h2, x, 10
Constant: a monomial that is a real number. Example: 25
Multiplying and Dividing MonomialsBut what is a monomial?
MultDivideMonomials.notebook
7
September 18, 2014
Monomials: a number, a variable, or product of a number and one or more variable(s) with nonnegative integer exponents.
Example: h2, x, 10
Constant: a monomial that is a real number. Example: 25
17 c 8f2g 34
5t
Multiplying and Dividing MonomialsBut what is a monomial?
MultDivideMonomials.notebook
8
September 18, 2014
Monomials: a number, a variable, or product of a number and one or more variable(s) with nonnegative integer exponents.
Example: h2, x, 10
Constant: a monomial that is a real number. Example: 25
17 c 8f2g 34
5t
No, the expression contains subtraction
Multiplying and Dividing MonomialsBut what is a monomial?
MultDivideMonomials.notebook
9
September 18, 2014
Monomials: a number, a variable, or product of a number and one or more variable(s) with nonnegative integer exponents.
Example: h2, x, 10
Constant: a monomial that is a real number. Example: 25
17 c 8f2g 34
5t
No, the expression contains subtraction
Yes, the expression is the PRODUCT of a number and more than a variable
Multiplying and Dividing MonomialsBut what is a monomial?
MultDivideMonomials.notebook
10
September 18, 2014
Monomials: a number, a variable, or product of a number and one or more variable(s) with nonnegative integer exponents.
Example: h2, x, 10
Constant: a monomial that is a real number. Example: 25
17 c 8f2g 34
5t
No, the expression contains subtraction
Yes, the expression is the PRODUCT of a number and more than a variable
Yes, the expression is a constant
Multiplying and Dividing MonomialsBut what is a monomial?
MultDivideMonomials.notebook
11
September 18, 2014
Monomials: a number, a variable, or product of a number and one or more variable(s) with nonnegative integer exponents.
Example: h2, x, 10
Constant: a monomial that is a real number. Example: 25
17 c 8f2g 34
5t
No, the expression contains subtraction
Yes, the expression is the PRODUCT of a number and more than a variable
Yes, the expression is a constant
No, variable can not be in the deniminator
Multiplying and Dividing MonomialsBut what is a monomial?
MultDivideMonomials.notebook
12
September 18, 2014
Aim: What are the Laws of Exponents for Multiplication and Division?
Just like we can multiply two numbers together, we can also multiply two exponents (or two monomials) together. To do this, we follow the Product of Powers Rule.
Product of Powers Rule: To multiply exponents that have the same base, simply add the exponents.
am • an = am+n
MultDivideMonomials.notebook
13
September 18, 2014
Aim: What are the Laws of Exponents for Multiplication and Division?
This can be seen if we actually count factors when multiplying exponents
75 • 74 =
It is very important to remember that bases have to beTHE SAME!!!!!!
= 79
9 factors
(7 • 7 • 7 • 7 • 7)
5 factors
(7 • 7• 7 • 7) •
4 factors
MultDivideMonomials.notebook
14
September 18, 2014
Examples1) 52 • 5 2) x3 • x5 3) w6 • w15
When there is a coefficient (number in front of the variable) we multiply them together.
4) 3x2 • 4x5 5) 2x5 • 3x7 6) 2a3 • (8a4)
MultDivideMonomials.notebook
15
September 18, 2014
Aim: What are the Laws of Exponents for Multiplication and Division?
Dividing exponents ... To divide exponents that have the same base, simply subtract the exponents. This is the Quotient of Powers Rule.
a9 ÷ a5 = a(9-5) = a4
a a a a a a a a aa a a a a
MultDivideMonomials.notebook
16
September 18, 2014
Examples7) f8 8) g52
When there is a coefficient, we divide them. 9) 12x5 10) (2)5 • 34 • 57 11) 15n7
f3 g2
2x2 (2)2 • 3 • 54 3n4
MultDivideMonomials.notebook
17
September 18, 2014
Homework ‐
WS p. 27 1‐10
11 & 12 are extra credit
MultDivideMonomials.notebook
18
September 18, 2014
MultDivideMonomials.notebook
19
September 18, 2014
MultDivideMonomials.notebook
20
September 18, 2014
Aim: How do we multiply a monomial with a polynomial? How do we divide a
polynomial by monomial?
MultDivideMonomials.notebook
21
September 18, 2014
Aim: How do we multiply a monomial with a polynomial? How do we divide a
polynomial by monomial?
To multiply a monomial by a polynomial:• Distribute• Follow the Laws of Exponents
1. Multiply the coefficients2. If the bases are the same keep the base and
add the exponents
2x(3x2 + 4x 6) =
MultDivideMonomials.notebook
22
September 18, 2014
Aim: How do we multiply a monomial with a polynomial? How do we divide a
polynomial by monomial?
To divide a polynomial by a monomial:• Divide each piece of the numerator by the denominator• Divide the coefficients• Follow the Laws of Exponents
1. Divide the coefficients2. If the bases are the same keep the base and subtract the exponents
Example: 25x 4 – 15x3 + 5x2 ______________________
5x 25x4 5x
______
5x______ 15x3
5x______+ 5x2
=
MultDivideMonomials.notebook
23
September 18, 2014
MultDivideMonomials.notebook
24
September 18, 2014
Aim: How do we simplify negative exponents?
**********************************************************************
Zero Exponents: Any quantity raised to a power of zero is equal to one.
Bonus: Why do zero exponents = 1?
MultDivideMonomials.notebook
25
September 18, 2014
Aim: How do we simplify negative exponents?
Negative Exponents: a number with a negative exponent is equal to the reciprocal of that power with a positive exponent.
* any base with a negative exponent is equal to 1 over the base with a positive exponent.
The base, 2, does not change. The negative exponent becomes positive -- in the denominator.
MultDivideMonomials.notebook
26
September 18, 2014
Aim: How do we simplify negative exponents?
To Multiply and Divide with negative exponents.
1) Follow the Laws of Exponents when multiplying and dividing
2) If your answer contains negative exponents, rewrite your answer using positive exponents
Example:
Problem After Division Simplified
MultDivideMonomials.notebook
27
September 18, 2014
MultDivideMonomials.notebook
28
September 18, 2014
MultDivideMonomials.notebook
29
September 18, 2014
MultDivideMonomials.notebook
30
September 18, 2014
Aim: How do we raise a power to a power?
Do Now:
MultDivideMonomials.notebook
31
September 18, 2014
Aim: How do we raise a power to a power?
MultDivideMonomials.notebook
32
September 18, 2014
MultDivideMonomials.notebook
33
September 18, 2014
Attachments
Fluency exponents in expanded form
Fluency Product Rule Numerals
Fluency Product Rule Variables
Fluency Quotient Rule
Fluency Quotient Rule Variables
Fluency power rule variables
Extra Credit Laws of exponents
