notes: exponent rules day 2...notes: exponent rules day 2 power of a product – 1. with numbers (2...
TRANSCRIPT
NOTES: EXPONENT RULES DAY 2
Topic Definition/Rule Example(s)
Multiplication (add exponents) xa ⋅ xb = xa+b
x 4 ⋅ x8 x5 y−2 ⋅ x 2 y
Power to a Power (multiply exponents)
xa( )b = xab x 3( )7
x −3( )−2
Power of a Product
(multiply exponents) ab( )n =anbn
x 4 y2( )3
3x 5( )4
Zero Exponents x0 =1
2ab( )0
2 ab( )0
Division (subtract exponents)
xa
xb = xa−b
x8
x2
x2
x8
Power of a Quotient
(multiply exponents)
xy
⎛
⎝⎜
⎞
⎠⎟
n= xn
yn
25
⎛
⎝⎜⎞
⎠⎟
3
2a4
b⎛
⎝⎜
⎞
⎠⎟
3
Simplifying
1. No Parentheses! 2. No Negatives! 3. Combine all like bases!
Before we deal with exponents that are VARIABLES and bases that are numbers, let's review exponents that are NUMBERS and bases that are variables……
Multiplying Powers –
1. With numbers 23 24 By definition Check it:
2. With numbers 34 31 By definition Check it:
3. With a partner -- Make one up with numbers – same base By definition Check it:
4. With VARIABLES – x6 x5 By definition
How about a rule:
Be careful with DIFFERENT BASES: x6 y5
The PRODUCT of POWERS
For the SAME BASES,
am an =
NOTES: EXPONENT RULES DAY 2 Power of a Product –
1. With numbers (2 3)2 By definition Check it:
2. With both (3 x)3 By definition Check it:
3. With a partner -- Make one up with numbers By definition Check it:
4. With VARIABLES – (x y)5 By definition
How about a rule:
Be careful with SUMS AND DIFFERENCES: (2 + 3)2 (6 - 2)3
(x +y)2
The POWER of PRODUCTS
(a b)n =
Power of a POWER –
1. With numbers (22)3 By definition Check it:
2. With numbers (32)4 By definition Check it:
3. With a partner -- Make one up with numbers By definition Check it:
4. With VARIABLES – (x3)5 By definition
How about a rule:
With a partner: Rewrite x16 as a) a product of powers b) a power of powers
The POWER of a POWER
(a m)n =
NOTES: EXPONENT RULES DAY 3 Powers of a Quotient –
1. With numbers
By definition Check it:
2. With numbers
By definition Check it:
3. With a partner -- Make one up with numbers By definition Check it:
4. With VARIABLES –
By definition
How about a rule:
The POWER of an QUOTIENT
For the Same Bases,
What do we do with negative exponents?
Power to a negative exponent –
1. With numbers 2-1 By definition Check it:
2. With numbers 3-4 By definition Check it:
3. With a partner -- Make one up with numbers By definition Check it:
4. With VARIABLES – x-5
By definition
How about a rule:
The POWER to a NEGATIVE Exponent
NOTES: NEGATIVE EXPONENT RULES DAY 3 I. Investigation
1. What does x-3 mean? ____________________________________________________________ 2. When might you come across a negative exponent?
II. Formula Definition: _________________________ Method: If there is a negative exponent:
- Move it to the denominator and make the exponent positive - Or move it to the numerator and make the exponent positive.
Ex 1: Ex 2:
Note: Think of a negative exponent as the reciprocal (________________________________ ). III. Practice Ex 3: Ex 4:
Ex 5: Ex 6: –12 x5 y–3 z–7
* Remember there is a difference between negative exponents and negative coefficients!
I. Multiplication with Fractions =)
Note A: Multiplying exponents with fractions is just like regular multiplication of fractions.
3.
x3
y9 i x8
y2 4. 12a i 1
4a2 5.
5a3b5
7⎛
⎝⎜⎜
⎞
⎠⎟⎟
14ab3
20⎛
⎝⎜⎜
⎞
⎠⎟⎟
II. Note B: True or False: A power is also distributed to the power of a coefficient. Note C: Exponents can only be combined with _______________________. 4. (5x3)6 5. (3x5)3 III. Negative Coefficient Note D: If a negative expression is raised to an even power, the answer is ____________________. Note E: If a negative expression is raised to an odd power, the answer is ____________________. 6. (-x)5 7. (-2y9)4 8. (-3y6)11 IV. Negative Exponents Rule: ____________________________ Example: _____________________ Note F: If you have a negative exponent, make it positive by taking the reciprocal of the base. Note G: A negative exponent can be moved to the numerator/denominator to become a positive exponent
14. 15. 16.
IV. Negative Exponent Combinations Step Method Example
1 Label all unlabeled exponents “1”
2 Take the reciprocal of the fraction and make the outside exponent positive.
3 Get rid of any inside parentheses.
4 Reduce any fractional coefficients.
5
Move all negatives either up or down. Make the exponents positive.
Subtract exponents of like bases. *Make sure the result is on the numerator!
6 Combine all like bases.
Move all negatives either up or down. Make the exponents positive.
7 Distribute the power to all exponents.
Power to a Zero Exponent –
1. With Numbers 30 By Definition: Check it:
2. With VARIABLES x0
By Definition
How about a rule:
The POWER to a ZERO Exponent
NOTES: EXPONENTIAL FUNCTIONS DAY 7 What functions have we studied so far?
Investigation: Complete these tables and sketch a graph of the function. Make observations about how x behaves and how f(x) behaves.
x f(x)= x + 2
0
1
2
3
4
5
6
7
8
9
x f(x)= x2
0
1
2
3
4
5
6
7
8
9
x f(x)= (2)x
0
1
2
3
4
5
6
7
8
9
Definition: An EXPONENTIAL FUNCTION is a NONLINEAR function
y = abx
with the following conditions: a ≠ 0, b ≠ 1, b › 0 Can the coefficient be negative? Can the base be negative? Can the exponent be negative? So exponential functions CAN look like this…… But not like this……. Can the base be a FRACTION? What happens then? Remember we have f(x) = bx
f(x) = 4x f(x) =
If b is more than one (b > 1), the exponential function is always: increasing decreasing If b is less than one (0 < b < 1), the exponential function is always: increasing decreasing Careful when you evaluate them! ONLY the base is raised to the exponent.
NOTES: EVALUATING EXPONENTIAL FUNCTIONS DAY 7 Evaluate the following for x = 3, x = 0, and then for x = -2 1. y = -2(5)x
x = 3
x = 0
x = -2
2. y = 3( )x
x = 3
x = 0
x = -2
3. y = 2(9)x
x = 3
x = 0
x = -2
4. y = 1.5(2)x
x = 3
x = 0
x = -2
NOTES: IDENTIFY EXPONENTIAL FUNCTIONS DAY 7 Which of the following ARE exponential functions? Explain.
A. x 0 1 2 3 y 4 8 16 32 B. x 0 1 2 3 y 2 4 6 8 C. x 0 1 2 3 y 8 4 2 1 D. x 0 1 2 3 y 0 1 8 27 Closing: With a partner, define YOUR OWN exponential function and complete the table. Specify the constant ratio. y = ____________________________
coefficient: __________
base: _______________
x 0 1 2 3 4 5 6
y
Final amount
Inital amount
Growth Rate
time
NOTES: EXPONENTIAL GROWTH AND DECAY DAY 8
Exponential Growth Functions
How does this compare to y = abx?
Ex.) 535 students attended the first Mr. Riverside contest. The attendance has
increased by 3% each year.
a. Write an exponential growth function to model the attendance of Mr. Riverside.
b. How many students will be attending in the 5th year?
y = a (1 + r) t
Final amount
Inital amount
Decay Rate
time
Exponential Decay Functions
Ex.) You just bought a car for $15,500. It depreciates 12% each year.
a. Write an exponential growth function to model the value of your car.
b. How much will your car be worth in 3 years?
y = a (1 - r) t
NOTES: EXPONENTIAL GROWTH AND DECAY DAY 8 Can you identify GROWTH and DECAY RATES? Be careful -- .2 is 20% and .02 is 2%.
Function Growth or Decay Rate (as a %)
1 y = 350(1 + .75)t
2 y = 575(1 – 0.6)t
3 y = 25(1.2)t
4 y = 240(.75)t
5 y = 12(1.05)t
6 y = 575(1 – .6)t
7 y = 25(1.2)t
8 y = 240(.75)t
Create the function described below:
1. A population of 210,000 increases by 12.5% each year
2. A $950 sound system depreciates at the rate of 9% each year
3. Milk costs $3.15/gallon and rises with the rate of inflation at 2% each year
4. A radioactive particle decays at the rate of 32% each year
NOTES: GRAPHING EXPONENTIAL FUNCTIONS DAY 8Transformations of exponential functions
y = abx – h + k
• a tells us the stretch or shrink • h tells us the horizontal shift
• k tells us the vertical shift
Graph the parent functions for exponential growth and decay:
x y = 2x –2 –1 0 1 2
Graph the following functions and state the domain and range.
A) f(x) = 3x+2 B) f(x) =
14
⎛⎝⎜
⎞⎠⎟
x−1
+2
x
–2 –1 0 1 2
Graphing Exponential Functions
1. Plot the points (0, 1) and (1, b). (for decay, it is easier to plot (0,1) and (–1, 1/b) )
2. Draw in the horizontal asymptote y = 0. 3. Fill in the curve.
PRACTICE: GRAPH EXPONENTIAL FUNCTIONS
3. f (x) = 2x−4 Asymptote: ______________________ Domain: ________________________ Range: __________________________ Zeros: ___________________________ Y-intercept: ______________________ Increasing: _______________________ Decreasing: _______________________
End Behavior:
As x → +∞ then f (x)→ _____As x → −∞ then f (x)→ _____
4. f (x) = 3 i 2x−5 +1
Asymptote: ______________________ Domain: ________________________ Range: __________________________ Zeros: ___________________________ Y-intercept: ______________________ Increasing: _______________________ Decreasing: _______________________
End Behavior:
As x → +∞ then f (x)→ _____As x → −∞ then f (x)→ _____
Function Domain Range Asymptote Equation
1 y = 2x
2 y = 3 i 2x+4
3 y = 2 i 3x −5
4 y = 3x+1
DAY 2: EXPONENT RULES
Topic Definition/Rule Example(s)
Multiplication (add exponents) xa ⋅ xb = xa+b
x 4 ⋅ x8 x5 y−2 ⋅ x 2 y
Power to a Power (multiply exponents)
xa( )b = xab x 3( )7
x −3( )−2
Power of a Product
(multiply exponents) ab( )n =anbn
x 4 y2( )3
3x 5( )4
Zero Exponents x0 =1
2ab( )0
2 ab( )0
Division (subtract exponents)
xa
xb = xa−b
x8
x2
x2
x8
Power of a Quotient
(multiply exponents)
xy
⎛
⎝⎜
⎞
⎠⎟
n= xn
yn
25
⎛
⎝⎜⎞
⎠⎟
3
2a4
b⎛
⎝⎜
⎞
⎠⎟
3
Simplifying
1. No Parentheses! 2. No Negatives! 3. Combine all like bases!
DAY 3: SIMPLIFYING EXPONENTS
Simplifying Exponents 1. Reduce any coefficients. 2. Get rid of any parentheses by distributing the exponents with multiplication. 3. Combine any like bases.
4x 3 y11w 3
8x 2 y10w 3
(−2x 3 y 5 )(4xyz 4 )
2x 0 y 3
12 y 4 • −4x 2
x 10z 4
Simplifying Negative Exponents 1. Reduce any coefficients. 2. Get rid of any parentheses by distributing the exponents with multiplication. 3. Combine any like bases. 4. Reciprocate any negative exponents. 5. Simplify!
6x4 y−9
12x−3 y−11
5 x3 (z2)4
15 (x−2)3 z5
⎛
⎝⎜⎞
⎠⎟
−2
DAY 3: RATIONAL EXPONENT OPERATIONS
Multiply with Rational Exponents 1. If the bases are the same, combine the exponents with addition. 2. Find a common denominator and add the numerators.
x14 • x
23
Power to a Power 1. Combine the exponents with multiplication. 2. No common denominator necessary!
x
15
⎛
⎝⎜⎞
⎠⎟
37
Divide with Rational Exponents 1. If the bases are the same, combine the exponents with subtraction. 2. Find a common denominator and add the numerators.
612
635
Divide with a Power 1. If the bases are the same, combine the exponents with subtraction. 2. Find a common denominator and add the numerators. 3. Then combine the exponents with multiplication. No common denominator necessary!
x2
5
x1
4
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
DAY 3: RATIONAL EXPONENTS Type Example Instructions/Notes
1 Negative Exponents
x−n = 1
xn 5−2 = Negative exponents
reciprocate!
2 Zero Exponent
x0 =1
490 =
5 x0⎛
⎝⎜
⎞
⎠⎟ =
Any expression to the 0th power is 1.
3
Convert Rational Exponents and Radicals
xmn = xmn
4932 =
125( )23 =
Power over Root
The denominator becomes the index of the
radical.
4 Multiplying Exponents
(with fractions) 532 •5
14
Add the exponents.
(common denominator)
5 Power to a Power (with
fractions)
x12
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
27
Multiply the exponents.
(no common denominator)
6 Dividing Exponents
(with fractions)
1125
1145
Subtract the exponents.
(common denominator)
7 Different Bases, Same
Exponents 314 •27
14
Multiply the bases. Combine the common
exponent into one exponent.
DAY 8: GRAPHING EXPONENTIAL FUNCTIONS 1. General Form: f (x )= a ⋅bx−h +k
2. Plot Key Points: 1st Point: (0, a) 2nd Point: (1, ab) or (–1, (b/a)) 3. Graph the horizontal asymptote: y = k. 4. The domain is all real numbers. The range is all real numbers (above or below k) Example 1: y = 3x+4 − 2
Asymptote: _____________________
Domain: ________________________
Range: _________________________
Example 2: y = 3(2)x−1+ 4
Asymptote: _____________________
Domain: ________________________
Range: _________________________
DAY 11: SIMPLIFYING RADICALS Simplifying Square Roots
1. Use the chart to find the largest perfect
square that divides evenly into the radicand (number under the radical)
2. Rewrite the radicand as a product (with
one of the factors as the number you just found)
3. Break up the radical into two (one with
the perfect square and one with the other factor).
4. Simplify the perfect square.
48
72
Simplifying Nth Roots
1. Use the chart to find the largest perfect power that divides evenly into the radicand (number under the radical)
2. Break up the radical into two (one with
the perfect power)
3. Simplify the perfect root.
2503
645
Simplifying a radical in the form: a b
1. Simplify the radical. 2. Multiply any like factors together.
3 12
DAY 15: SIMPLIFYING RADICALS
24x 4 y13
Simplifying Radicals with Variables
1. Find the perfect power that divides evenly into the coefficient.
2. Divide each exponent by the index
(root).
3. Break the radical into two (one that is a perfect root).
4. Simplify the perfect root.
54x 4 y133
DAY 13, 14: RADICAL OPERATIONS
Type Example
3 12
1
Multiplying Radicals (same root)
1. Multiply any coefficients. 2. Multiply the radicands. 3. Keep the same root. 2 103 ⋅ 5 43
2
Dividing Radicals (same root)
1. Divide the radicands. 2. Keep the same root.
3 nth Root of a to the nth power
The root and the power cancel out!
x77
6
Add and Subtract Radicals
1. Radicals can only be combined with addition/subtraction if they have the same radicand.
2. Simplify each radical separately. 3. Combine the coefficients. 4. Keep the same radical.
32 + 3 2
DAY 14: RATIONALIZE THE DENOMINATOR
Single Term Denominator
3
5
10
3 2
Denominator with Two Terms
7
1+ 2
5
7+ 3 3
Nth Root Denominator
4
73
8
53
DAY 16: COMPLEX NUMBER OPERATIONS ADD/SUBTRACT COMPLEX NUMBERS
1. Get rid of the parentheses by distributing
any negatives. 2. Combine like terms. 3. Write in the form a + bi.
(2 + i) – (6 – 5i)
MULTIPLY COMPLEX NUMBERS
1. Distribute the binomials (FOIL). 2. Recall: i2 = –1 3. Combine like terms. 4. Write in the form a + bi
(2 + i) (6 – 5i)
DIVIDE COMPLEX NUMBERS
1. Multiply the numerator and denominator by the complex conjugate of the denominator (exactly the same as the denominator except the sign).
2. Distribute the binomials (FOIL). 3. Combine like terms. 4. Simplify.
3+ 2i2+ i
DAY 16: IMAGINARY NUMBER OPERATIONS
DEFINITION OF i i = −1 i2 = –1
POWERS OF i If the i has an exponent that is a multiple of 4,
the answer is 1.
imultiple of 4 = 1 imultiple of 4 + 1 = –1 imultiple of 4 + 2 = i imultiple of 4 + 3 = –i
i47
−36
−52
−16x 8
SIMPLIFYING RADICALS WITH NEGATIVES
1. A negative in the radical can become i
outside the radical.
2. Simplify the radical.
−16x 83
RADICAL OPERATIONS
1. Simplify the imaginary part of the
radicals by bringing i outside the radical.
2. Simplify the radicals.
3. Add/Subtract any like radicals.
−8 + −18
