8.1 Multiplication Properties of Exponents Multiplying Monomials and Raising Monomials to Powers Objective: Use properties of exponents to multiply exponential expressions.

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 nonzero number a, and all integers m and n, a m a n = a m+n. Product of Powers Rule Notice that the Produce of Power Rule is only for those powers of the SAME BASE

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 nonzero 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 Product rule. (ab) m = a m b m

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

Summary Remember the properties of exponents. 1. Product of Power Rule For any nonzero number a, and all integers m and n, 2. Power of Power Rule For any nonzero number, a, and all integers m and n, 3. Power of Product Rule For any nonzero numbers, a and b, and integer m,

Guided Practice – L 8.1 DHQ PAGE: 453 in Textbook #’s 4-20 EVEN ONLY

Assignment – Lesson 8.1 P. 453 #’s 22-31, 34-45, 52-60

Chapter 8 Exponents Section 8.2 Zero and Negative Exponents

y y (-5) (-5)

x 3(2) x6x6 5 2(3) 5656 (-2) 3(4) (-2) (a – 2) 3(2) (a – 2) 6

3 4 x 4 y 4 (-3) 2 y 2 9y y 2 -9y x 8 y 2 y 5 9x 8 y 7

3737 z7z y x 16 y 5

Undefined

(-2)(-3) 5656

g 2n

6 0 = 1

Summary Remember the properties of zero and negative exponents 4. Zero Exponent Rule For any nonzero number a, 5. Negative Exponent Rule For any nonzero number, a, and all integer n,

Guided Practice – L 8.2 DHQ PAGE: 459 in Textbook #’s 3-11 ODD ONLY

Assignment – Lesson 8.2 P #’s EVEN ONLY + #’s ALL

Chapter 8 Exponents Division (Quotient) Properties of Exponents

Notice that when answering the questions, you have to combine all the rules you have learned, not just the rules you learned in this section.

= 1

=10 3 = a 5

Power of Product, Power of Power

a. = (x 5 – 3 ) -1 = (x 2 ) -1 = x -2 Remember! Simplify the inside expression first!!!

-4m 4 n 2

Summary 1.The division property of the exponents is only for the power of same base. 2.The other learned properties may be used at the same time.

Summary Remember the properties of Quotient of Power and Power of Quotient. 6. Quotient of Power Rule For any nonzero number a, and all integers m and n, 7. Power of Quotient Rule For any nonzero number, a, b, and all integer m,

Guided Practice – L 8.3 DHQ PAGE: 466 in Textbook #’s 3-5, 11-13

Assignment – Lesson 8.3 P. 466 #’s ALL

Objectives 1.Write and use the models for exponential growth. 2.Write and use the models for exponential decay.

Exponential Growth Functions 8.5 E XPONENTIAL G ROWTH M ODEL A quantity is growing exponentially if it increases by the same percent in each time period. C is the initial amount. t is the time period. (1 + r) is the growth factor, r is the growth rate. The percent of increase is 100r. y = C (1 + r) t Sometimes use P instead of C Note: measure of rate and time MUST be in the same time unit

Example 1 Compound Interest You deposit $1500 in an account that pays 2.3% interest compounded yearly, 1)What was the initial principal ( P ) invested? 2)What is the growth rate ( r )? The growth factor? 3)Using the equation A = P(1+r) t, how much money would you have after 2 years if you didn’t deposit any more money? C or P = $1500 growth rate (r) is The growth factor is y = $

1.What is the percent increase each year? 2.Write a model for the number of rabbits in any given year. 3.Find the number of rabbits after 5 years. Example 2 Exponential Growth Model A population of 20 rabbits is released into a wild- life region. The population triples each year for 5 years. 200% ≈ 4860 rabbits R =20(1+2.00) t

Example 2 Exponential Growth Model Graph the growth of the rabbits. Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through the points. t P Time (years) Population P = 20 ( 3 ) t Here, the large growth factor of 3 corresponds to a rapid increase

E XPONENTIAL D ECAY M ODEL C is the initial amount. t is the time period. (1 – r ) is the decay factor, r is the decay rate. The percent of decrease is 100r. y = C (1 – r) t Exponential Decay Functions 8.6 A quantity is decreasing exponentially if it decreases by the same percent in each time period. Sometimes use P instead of C Note: measure of rate and time MUST be in the same time unit

Example 4 Asset Depreciation You buy a new car at $22,500. The car depreciates at the rate of 7% per year. 1.What was the initial amount invested? 2.What is the decay rate? The decay factor? 3.What will the car be worth after the first year? The second year? C or P = $22,500 growth rate (r) is The growth factor is.93 y 1 = $20,925y 2 = $19,460.25

E XPONENTIAL G ROWTH AND D ECAY M ODELS y = C (1 – r) t y = C (1 + r) t E XPONENTIAL G ROWTH M ODEL E XPONENTIAL D ECAY M ODEL 1 + r > 1 0 < 1 – r < 1 C ONCEPT S UMMARY An exponential model y = a b t represents exponential growth if b > 1 and exponential decay if 0 < b < 1. C is the initial amount.t is the time period. (1 – r) is the decay factor, r is the decay rate. (1 + r) is the growth factor, r is the growth rate. (0, C)

Example 6 Exponential Growth or Decay Your business had a profit of $25,000 in If the profit increased by 12% each year, what would your expected profit be in the year 2010? Identify C, t, r, and the growth factor. Write down the equation you would use and solve. y = $97,399.40

Example 6 Exponential Growth or Decay Iodine-131 is a radioactive isotope used in medicine. Its decay rate is 50%. If a patient is given 25mg of iodine-131, how much would be left after 32 days or 4 half-lives. Identify C, t, r, and the decay factor. Write down the equation you would use and solve. y = 1.56 mg

The graph of exponential growth function is C The graph of exponential growth function is like an air plane taking off.

The graph of exponential decay function is C The graph of exponential decay function is like an air plane landing.

G RAPHING E XPONENTIAL D ECAY M ODELS E XPONENTIAL G ROWTH AND D ECAY M ODELS y = C (1 – r) t y = C (1 + r) t E XPONENTIAL G ROWTH M ODEL E XPONENTIAL D ECAY M ODEL 1 + r > 1 0 < 1 – r < 1 C ONCEPT S UMMARY An exponential model y = a b t represents exponential growth if b > 1 and exponential decay if 0 < b < 1. C is the initial amount.t is the time period. (1 – r) is the decay factor, r is the decay rate. (1 + r) is the growth factor, r is the growth rate. (0, C)

1.Exponential growth and decay models are y = C(1 + r) t y = C(1 – r) t Or they can be combined to one: y = C(1 ± r) t 2.Their graphs are always pass through point (0, C). 3.The graph of exponential growth functions is strictly increasing. The graph of exponential decay functions is strictly decreasing. Their graphs always stay above the x- axis.

Guided Practice – L 8.5/8.6 DHQ PAGE: 480 in Textbook # 4 PAGE: 488 in Textbook # 5 PLEASE SHOW WORK!

Assignment – Lesson P. 480 #’s 6-11 P. 488 #’s SHOW WORK!