Created by K.Snyder 1

HONOR-A2-TRIG UNIT 7

EXPONENT LAWS (DAY 1)

Warm-Up: Find the exact roots of the equation: 0128x16x8x 23

Simplify each of the following.

1) (3d3)4 2) (4xy4z2)(3x6y9z2) 3) 3

4

xy

)xy(

4) 3

3

3

6= 5)

k

2k

x

x

6)

5

c

c4

y

y

EXPONENT LAWS SIMPLIFY MULTIPLICATION LAW

(w/same bases)

Rule: __________exponents

75 xx = )xy5)(yx2( 52 =

DIVISION LAW:

(w/same bases)

Rule: ____________ exponents

5

8

a

a

x

3x

g

g

EXPONENT RAISED TO EXPONENT

(POWER LAW)

Rule: _____________exponents

43)x( = 326 )yx5( =

4

3

2

x5

y3

ZERO EXPONENTS

Rule: Anything to the zero = ___

0)x3( =

032 )xa6( = 0b5 =

Created by K.Snyder 2

7) (3a2b3c-4) (5ab2c6) 8) 107

1054

zxy7

zyx35

9) (5x2y6)5 10) 2a

4a

x

x

11)

3

x2

x8

y

y

12)

02

xy5

xy10

13) -40 14) (-4)0

15) (2x)0 16) 2x0

Created by K.Snyder 3

NEGATIVE EXPONENTS (DAY 2) Simplify:

Using Division Rule for Exponents: 5

3

x

x V.S. Canceling Method:

5

3

x

x

What is the relationship between this answer and the one above?

STEPS FOR REWRITING NEGATIVE EXPONENTS INTO POSITIVE EXPONENTS:

Example: ab-n = nb

a

RE-WRITE USING ONLY POSITIVE EXPONENTS:

1. 7y-1 2. 6c

a

3. 3

2

y

x

4. (7k)-3

5. 23

2

ca

c2

6. )ba5)((ba3( 9342 7. d6

d3 5

8. 63

32

yx

y)x(

9. Using only positive exponents, write an equivalent expression to 2

4

y

x5

.

10. The expression 21 )a3)(a9( is equivalent to

(1) 3a (2) 3a81 (3) 3a27 (4) a27

11) If f(x) = 2x0 + 3x-3, find f(5). 12) If f(x) =

2

2

1

)x2(

x

, find f(4)

Created by K.Snyder 4

FRACTIONAL EXPONENTS

Recall: Simplify: 4 53yx

Re-write radical form using fractional exponents:

(1) x5 = (2) 3x = (3) 4 x = (4) 3 2x =

(5) 3 yx = (6)

3yx = (7) 2 5 b = (8) x

x22 3

Rewrite fractional exponents expression into radical form.

(9) 3

2

125 (10) 2

10 33

(11) 3

2

216

(12) 5

3

32

(13) 2

1

a3 = (14) 41

ab = (15) 43

x (16) 6

5

x4

)(

17) If 2

3

xxf

)( , find f(16). 18) Evaluate 23

1

0 aaa when a = 8.

Radical Form to Fractional Exponent Form

n ax or an x to n

a

x

Created by K.Snyder 5

SOLVING FRACTIONAL EXPONENT AND RADICAL EQUATIONS (DAY 3)

1. Solve for x: x3 = 8 2. Solve for y: 6y2 3

1

3. Solve for w: 91w 2

3

4. Solve for x: 93x 5

2

5. Solve for a: 16

12a32 2

5

6. Solve for x: 3112z5 3

5

7. Solve for x: 4x85 8. Solve for x: 22 + 212c7

9. Solve for a: 1250a2 4 10. Solve for x: 3072x4

3 6

PROCEDURE FOR FRACTIONAL EXPONENT EQUATIONS:

1. Get variable with fractional exponent by itself.

2. RAISE each side to the reciprocal exponent. (This eliminates the exponents)

If the reciprocal power has an EVEN “ROOT” number in the denominator

there will be two answers ( ).

3. Solve remaining equation

4. Check answer in ORIGINAL EQUATION

If not a fractional Exponent: take the ROOT (#) of this equation on both sides and solve.

Created by K.Snyder 6

SOLVING EXPONENTIAL EQUATIONS USING SAME AND DIFFERENT BASE(DAY 4)

Write the following with the lowest possible integral base.

1) 25 = 2) 32 = 3) 25

1=

Solve the following:

1) 32x 44 2) x32x 55 3) 77 4x

4) 9x = 27 5) 36

1= 62x 6) 255 3x

7) 4x+1 = 8x 8) 23y-6 = 8 9) xx

184

1

10) 2 4xx2

11) 3x2x 6416 12) x1x 343)49

1(

PROCEDURE FOR SOLVING EXPONENTIAL EQUATIONS W/EXPONENTS ON BOTH SIDES OF EQU)

1. MUST HAVE SAME BASES!!!

2. Set the exponents equal to each other and solve for x.

3. Check your answer in the original equation.

Created by K.Snyder 7

GRAPHING EXPONENTIAL FUNCTIONS (DAY 5)

1) Graph y = 3x. Create a table of values showing at least 5 points.

Domain:

Range:

2) Graph y =

x

3

1

. Create a table of values showing at least 5 points.

Domain:

Range:

Created by K.Snyder 8

EXPONENTIAL GROWTH AND DECAY

y = abx

PROPERTIES TO KNOW:

1) “b” is called the______________________ It is always a ___________________

2) “a” is the __________________. If a = 1, the point _________ is on the graph.

3) The domain is always:

4) The range is always:

5) The graph NEVER _____________________________________________________

6) The x-axis (y = 0) is an ______________________ of the graph.

TYPES OF GROWTH:

If “b” is __________________, you have exponential __________

o The graph __________________

.

If “b” is___________________________________you have

exponential __________

o The graph __________________

An exponential growth curve graphed with its

exponential decay curve is a reflection over the y-axis.

WHEN WORKING WITH APPLICATIONS:

1) “a” is the ____________________ , population, number, etc. It is your _______________.

2) “b” is the___________________. (always turn % into a decimal)

3) “t” or “x” usually refers to time, unless otherwise stated.

x3y

x

x

2

1y

or

2y

Created by K.Snyder 9

PROCEDURE FOR APPLICATION PROBLEMS

1. BOX equation if given. Determine if it is an exponential function problem

Plug in information given and solve for missing parts

IF NOT GIVEN AN EQUATION in the word problem

1. Look for clue words that describe an exponential function.

2. Determine initial value (a #)

3. Determine the growth rate (b #) as a decimal

4. Write equation

5. Plug in information given and solve for missing parts

1) In the 2000-2001 school year, the average cost for one year at a four-year college

was $16, 332 which was an increase of 5.2% from the previous year. If this trend

were to continue, the equation C(x) = 16, 332(1.052)x could be used to model the

cost, C(x), of a college education x years from 2000.

a) Find C(4). What does this number represent?

b) If this trend continues, how much would parents expect to pay for their new

born baby’s first year of college? (Assume the child would enter college in

18 years.)

2) Given the function F(x) = 20(0.90)x, where F(x) is the number of fish in Jason’s fish

tank and x is the number of days since Jason set up the tank.

a) How many fish did Jason have to start?

b) How many fish does Jason have after 2 days?

c) What is happening to Jason’s fish? Explain why.

Created by K.Snyder 10

EXPONENTIAL APPLICATIONS CONT… (DAY 6) RECALL:

1) In 1993, the population of New Zealand was 3,424,000, with an average annual

growth rate of 1.3%. Suppose that this growth rate were to continue.

a) Express the population P as a function of n, the number of years after 1993.

b) Estimate New Zealand’s population in the year 2010.

2) There are 2 kilograms of a certain substance and it decays by 10% each year.

a) After n years, how much of the substance would remain?

b) How much of the substance would remain after 4 years?

c) After six years would more than half of the substance decayed?

3) If Brent bought a new car for $23,545 in 2004 and it depreciates by 12.5% each

year. How much will the car be worth in 2011?

4) The population of Tanzania in 1995 was about 28.5 million, with an annual growth

rate of 3%. Predict what the population will be in the year 2008.

Created by K.Snyder 11

5) Atmospheric pressure starts at 14.7 pounds per square inch at sea level, and it goes

down 19% for each mile you are above sea level.

a) Write an exponential equation representing atmospheric pressure as a

function of height h in miles above sea level.

b) What would the atmospheric pressure be to the nearest tenth at an altitude

of 6.25 miles.

6) Graph f(x) = 2-x over the domain of –3 < x < 3.

a) Is this an example of exponential growth of decay?

b) Graph the reflection of f(x) over the y-axis.

c) Write the equation of the reflection.

Created by K.Snyder 12

COMPOUND INTEREST (DAY 7)

Most banks compound interest more than once a year.

Annually = _______ time per year Quarterly = _______ times per year

Semi-annually = _______ times per year Monthly = _______ times per year

When interest is compounded a certain number of times per year (n) such as

monthly or quarterly, use the formula, where:

A = _____________________ r = _____________________

P = _____________________ t = _____________________

n = _____________________

If compounding takes place without interruption, called compounded

__________________________, use the formula, where:

e = _____________________

Problems:

1) Sara had invested $800 in a savings account that paid 4.2% interest compounded

annually. How much money was in the account after 4 years, if he left the money

untouched?

Compounding Interest (n) Times a Year:

A P 1r

n

(nt)

Compounding Interest Continuously:

A Pert

Created by K.Snyder 13

2) Mrs. Capizzi has put $2,500 in a 5-year CD (certificate of deposit) that pays 7.4%

compounded quarterly. To the nearest dollar, how much will the CD be worth

when it matures?

3) When their daughter was 2 years old, the McGinn’s paid $7,000 for a 15-year

college bond for her. The bond pays 7.1% compounded monthly. Their banker told

them that if the money were left alone, it would triple in value. Is the banker right?

4) If Bailey invested money 4 years ago with an annual interest rate of 3.275%

compounded monthly, and it is valued at $11,260, how much money did she

initially invest?

5) Susie invested $2,000 at a rate of 3.5% per year, compounded continuously. How

much will she have after 10 years?

Created by K.Snyder 14

6) Nancy has $4,000 in a savings account that pays 3.9% interest compounded semi-

annually. Suppose the money is left untouched for 10 years.

a. How much money is in the account b. How much money is left in

after the first five years? The account after 10 years?

c. Does the account earn more interest during the first five years ir the second

five years? Explain why this is so.

7) Branden invested $1,800 in an account that pays 4.4% compounded daily (365

days in a year). If he leaves the money alone, how much will be in the account

after 2.5 years?

8) If Brent’s account is worth $5300 in the year 2005 and the account pays an annual

interest rate of 4.3%, compounded weekly. How much did Brent have to put in in

the year 1998 to have this much money now?

9) In a state park, the deer population was estimated to be 2,000 and increasing

continuously at a rate of 4% per year. If the increase continues at that rate, what is the

expected deer population in 10 years?