Algebra 1 Unit 5 Working with Exponents
Name ___________________________
Per ______
Objective 1: Simplifying Expressions with Exponents (Powers INSIDE parentheses) Exponents are a shortcut. They are a quicker way of writing repeated multiplication. How to simplify exponential expressions 1) Expand all exponents. Since exponents tell us how many of a certain letter are being multiplied to itself, we “expand” out the problem so we can COUNT how many of each letter there are. .  2) Use exponents to "snap back". • Multiply all numbers. • Multiply all letters by using the exponent. Wrote the letter ONCE with the correct exponent next to it. 3) Anything raised to the power of 0 will simplify to 1 An exponent on a single variable or number means to write out a multiplication problem with that variable or number multiplied to itself. Write out the variable or number as many times as it says to in the exponent. Note: If the variable has a negative sign in front of it, there is a HIDDEN -­‐1 that is multiplied to the power. You do NOT write out the -­‐1. Write a negative sign as "times negative 1" Examples: Expand: Ex 1: x 5 Ex 4:
-2n5
Ex 2: 34x5 Ex 5: -n6
Ex 3: (2x3) y5
Objective 1: Quick Practice
Simplify the following by first expanding and then snapping back:
1) n 2 i n 7
3) −x 3 (−x 4 )
5) (−4x 3 )(−5x 7 )
2) n(n 2 )(n 4 )
4) (2x 2 )(8x)
2 4
3 6
6) −4(2x y )(5x y )
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Objective 2: Simplifying Exponential expressions: Powers OUTSIDE parentheses
•
•
Simplify inside the parentheses as much as possible before continuing the problem. If there is a power outside parentheses, write out the ENTIRE part in the parentheses times itself. Write it as many times as it says to in the exponent outside the parentheses. This is a required step. Examples: Expand and simplify:
Ex 1) (-n)6
Ex 2) -(n)6
Ex 3) (-n)5
Ex 4) -(n)5
Ex 5) (-5x4y)(2xy2)3
Objective 2: Quick Practice
Simplify the following:
1)
(x )
2)
(x ) (x )
5 2
2 5
= x5 i x5 =
3
3)
( 4x y )
4)
( −2x )3 ( 3y 5 )
2
3
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5) (−x)3 (3xy 2 )2
6)
n(n2)5n3
Practice: Objective 1
1. n(n 5 )
6. n i n 6 i n 0
2. y(y5)
7. (-x3)(-x4)
3. n2 • n7
8. (2a2)(8a)
3
4
12. (−7x )(x )
9. (x2y)(4xy3)
4. (-7x2)(x4)
10. (−7x 3 )(x 4 )
5. x(x2)(x4)
11. n i n 6 i n 0
13. (−x 2 y)(4xy 3 )
2 4
6
14. (−3x y )(2xy )
15. x0
Practice: Objective 2
1)
(x )
7) -2(2x4)2(5x3)
4) (−n)4 (−2n)3
5 2
2) (−n)4 (−n 3 )
5)
( −3mn )
3) −3( mn 4 )
6)
( 4 x y ) ( 2y )
5
4 3
2
2 3
5
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Objective 3: Dividing with Exponents
Important Fact:
• Common letters on different sides of a fraction bar will CANCEL to 1's. (Make sure extra letters stay in the correct place by fraction bar) • Numbers will also divide or common factors will cancel to 1's (Make sure numbers stay in the correct place by fraction bar!) −2x
−2x
Ex 1)
10x
10
Ex 2)
5
5x
Ex 3)
Ex 4)
6
2x
Ex 5)
2
6x
2x
Ex 6) 6
Objective 3 Quick Practice:
Expand each expression and then snap it back together using exponents.
1.
m6
m4
4.
6a 2
3a 5
7.
a 5b 4
a 2b
15m 4
2.
10m 6
3a 5
5.
6a 2
14a 2
8.
2a
12a 2
3.
18a 5
6a 5
6.
3a 2
3xy 6
9.
12y 4 x
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Practice: Dividing with Exponents Page 5
Practice (mixed problems)
Expand each expression and then snap it back together using exponents.
1. (2a2b3)(-3ab4)
5.
12 ab
6a 2 b 3
2. (2a2b3)2
10. (a2b4)(a2)(b2)
6.
3.
m
m3
2a 2 b 3
3ab 4
m3
7.
m
4.
9. x(x2)(x7)
12 a 2 b 3
6ab
− 21w 5 x 2
8.
7w4 x 5
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Exponent Bingo
Free Space!!!
Bingo Problems:
1)
6)
11)
2)
7)
12)
3)
8)
13)
4)
9)
14)
5)
10)
15)
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Objective 4: Dividing with Exponents and powers OUTSIDE parentheses
•
•
•
Simplify inside the parentheses as much as possible before continuing the problem. If there is a power outside parentheses, write out the ENTIRE part in the parentheses times itself. Write it as many times as it says to in the exponent outside the parentheses. This is a required step. You may answer the rest of the problem without further expanding if you wish. ⎛ 12x 4 y 5 ⎞
Ex 1) ⎜
⎝ 4 x 2 y 2 ⎟⎠
⎛ 5x 5 y ⎞
Ex 3) ⎜ −
⎝ 15x 3 ⎟⎠
3
⎛ 3x 6 y 3 ⎞
ex 2) ⎜ − 5 ⎟
⎝ 2x y ⎠
3
Ex 4)
⎛ 6abc 0 ⎞
⎜⎝ 5b 3 ⎟⎠
Objective 4 Quick Practice
⎛ 2x 2 y 2 ⎞
1) ⎜
⎝ 8x ⎟⎠
⎛ 2r 5 w ⎞
3) ⎜ 4 3 ⎟
⎝r w ⎠
4
3
⎛ 12x 0 y 6 ⎞
2) ⎜
⎝ 3y 3 ⎟⎠
⎛ xy 7 ⎞
4) ⎜
⎝ 3xy 2 ⎟⎠
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4
2
2
2
Objective 5: Negative Exponents
Important fact:
Negative Exponents are "position bumpers" of the entire power relative to fraction bar. To simplify expressions with negative exponents:   1) Make sure there is a fraction bar. (if there is no fraction bar, create one under the problem 2) Move all powers with negative exponents to their correct position relative to the fraction
3) FULLY simplify expressions INSIDE the ( ) if possible. 4) ALWAYS EXPAND if there is a power outside the parentheses. Rewrite the following without negative exponents
Ex 1: 5a3b-4
Ex 2) -3x-4
Ex 3)
Objective 5: Quick Practice 1)
m −4
m
m
3)
m −4
5)
(−x −1 y)0
4w −1 y 2
x −4 y 0
7)
x2
2)
p −8
p3
(a b )
2
4)
6)
3 2
( ab )−2
x −4
x −5
⎛ 4m 2 n 2 ⎞
8) ⎜
⎝ 8m −1n ⎟⎠
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0
Practicing all types
Hidden Message Card 1’s Letter ________ Copy and solve below
Card 5’s Letter ________ Copy and solve below
Card 2’s Letter ________ Copy and solve below
Card 6’s Letter ________ Copy and solve below
Card 3’s Letter ________ Copy and solve below
Card 7’s Letter ________ Copy and solve below
Card 4’s Letter ________ Copy and solve below
Card 8’s Letter ________ Copy and solve below
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Card 9’s Letter ________ Copy and solve below
Card 13’s Letter ________ Copy and solve below
Card 10’s Letter ________ Copy and solve below Card 14’s Letter ________ Copy and solve below
Card 11’s Letter ________ Copy and solve below
Card 15’s Letter ________ Copy and solve below
Card 12’s Letter ________ Copy and solve below Card 16’s Letter ________
Hidden Message: ______________________________________________________
Hidden Message ____________________________________________________________________
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More Practice Problems
Set #1: Simplify the following completely. All final expressions should only have positive exponents.
2
1) (4x)(−3xy ) 3
4) (−2x) (−x) (−4xy 2 )2
5
7) 8x y 5 2
2) 3x(2xy)(−3x y ) 2
5) (4x)(−3xy ) −3xy 7
3 5
3) 6x y 9x 2
−3
6) 18y ( 5x y )
0
8)
3 −2
−12x −2 y −3
−7 5
9) 6x y Set #2: Expand each expression and then snap it back together using exponents.
1.
m
m4
5.
5x 2 y
x3
9.
10a −2 b 3
5a 4 b
2.
m
m −4
6.
5 x −2 y
x3
10.
10a 2 b −3
5a 4 b
3.
p −8
p3
b −4
4.
b −5
⎛ 5x 2 y ⎞
7. ⎜⎜ −3 ⎟⎟
⎝ x ⎠
8.
2
10 a 2 b 3
5a 4 b
Page 12
⎛ 10a 2 b 3 ⎞
11. ⎜⎜ −4 −1 ⎟⎟
⎝ 5a b ⎠
3
Objective 6: Scientific Notation
Scientific notation is a format used to write very large and very small numbers. It saves space and
makes comparing numbers easier. It is often used in science for long distances or small sizes (atoms)
There are 3 parts to a number in scientific notation.
Part 1: A number in scientific notation starts with a number between 1 and 10 (can be 1, can’t be 10).
Part 2: A multiplication symbol is next. Usually × is used but sometimes • or ∗ .
Part 3: The number 10 to some power is last. The power can be any integer (…-2,-1, 0, 1, 2, …)
Format: c × 10 n Notice the old multiplication symbol is used here. This is very common although
again, a dot or asterisk may also be used. c × 10 n or c ∗ 10 n or c • 10 n
Numbers in scientific notation
2 × 10 3
5.679 × 10 −2
1× 10 25
9.33958827 × 10 −5
Numbers NOT in scientific notation (why NOT on the blank)
0.134 × 10 3 ________________________________
________________________________
− 3× 10 4
−5
________________________________
10 × 10
15
________________________________
265× 10
NOTE: Calculators (not people) often write numbers in scientific notation like this: cE n with a lot of
decimal numbers in c. 5.2 E 23 means 5.2 × 10 23 . Put 5.2 × 10 23 in your calculator and press ENTER.
POSITIVE POWERS OF 10
If the exponent on the 10 is positive, the number represented in scientific notation is very LARGE.
How do I know this? The first number is multiplied over and over by 10, moving the decimal point one
place to the RIGHT for every exponent value on 10.
Example 1: Change 4.53 × 1011 from scientific notation to decimal form. This is a large number.
Answer: The number 4.53 will be multiplied by 10 five times, becoming 453,000,000,000.
Rewrite the following numbers from scientific notation to decimal (standard) form.
1. 3.4578 × 10 9 _________________________
3. 1.3 × 10 0
_________________________
2. 8.497 × 1013 _________________________
4. 9 × 10 5
_________________________
If a number is very large, it can be put in scientific notation,
Example 2: Put 53,419 into scientific notation.
Answer: The exponent on 10 will be positive since 53,419 is a large number. Find the spot where the
decimal point should be so the first number will be between 1 and 10. The spot is between 5 and 3.
Rewrite 5.3419 × 10 4 . Notice the exponent is positive to indicate a large number.
Put the following numbers into scientific notation.
1) 36.41
3) 718,569
____________________
____________________
2) 4,562.23
____________________
4) 9.32
____________________
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Scientific Notation and NEGATIVE powers of 10
If the exponent on the 10 is negative, the number represented in scientific notation is very SMALL.
How do I know this? The first number is divided over and over by 10, moving the decimal point one
place to the LEFT for every exponent value on 10.
Example 1: Change 2.64 × 10 −7 from scientific notation to decimal form. This is a small number.
Answer: The number 2.64 will be divided by 10 seven times, becoming 0.000000264
Rewrite the following numbers from scientific notation to decimal (standard) form.
1. 7.48 × 10 −9 _________________________
3. 1.3 × 10 −3
_________________________
2. 4.567 × 10 −14 _________________________
4. 9 × 10 −5
_________________________
If a number is very small, it can be put in scientific notation,
Example 2: Put 0.000 000 459 into scientific notation.
Answer: The exponent on 10 will be negative since 0.000 000 459 is a small number. Find the spot
where the decimal point should be so the first number will be between 1 and 10. The spot is between 4
and 5. Rewrite 4.59 × 10 −8 . Notice the exponent is negative to indicate a large number.
Put the following numbers into scientific notation.
1) 0.000 000 000 376 5
3) 0.0056
____________________
____________________
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2) .4523
____________________
4) 3
____________________
Practice Objective 6
Express each number in standard form.
1. 4.91 × 104
4. 5 × 10-3
7. 1.7087 × 107
2. 4.91 × 10-4
5. 2.001 × 106
8. 3.5 × 10-2
3. 5 × 103
6. 2.001 × 10-6
Express each number in scientific notation.
9. 5,100,000
14. 0.002002
10. 80,300,000,000
15. 68,070,000,000,000
11. 0.0049
16. 0.17
12. 14,000
17. 0.00006
13. 901,050,000,000
18. 800,000,000
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Objective 7: Applying Exponents: Exponential Growth and Decay
y = the final amount
a = the initial amount
Exponential Growth
r = rate of growth/decay as a
decimalby a percentage rate over time. If anything grows at a
Money in banks and populations often increase
t = time passed
percentage rate, r%, then for every time period that passes, the item is worth (100% + r%) of its
previous value. The exponent stands for time that passes.
Exponential Growth:
y = a(1 + r)t
Example 1: $1500 is deposited in a savings account paying an annual yield of 6%.
a) What is the percent written as a decimal? ____________
b) Write an exponential growth equation.
c) If the account is left alone, how much money will be in the account at the end of 10 years?
Example 2) Jaime invests $100 at an annual yield of 5%.
a) What is the percent written as a decimal? ____________
b) Write an exponential growth equation.
c) If the account is left alone, how much money will be in the account at the end of 10 years?
Example 3: A home is worth $125,000. It is appreciating at a rate of 3% per year.
a) What is the percent written as a decimal? ____________
b) Write an exponential growth equation.
c) How much money will the house be worth in 5 years?
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y = the final amount
a = the initial amount
r = rate of growth/decay
Automobiles and other manufactured goods
t =often
time decrease
passed in value over time. This decrease is called
depreciation. If anything decreases at r%, then each time period the item is worth (100%-r%) of its
Exponential Decay and Depreciation
previous value.
Exponential Decay:
y = a(1 – r)t
Example 1
In 2010, a new Firebird Trans Am cost $15,798. Suppose the car depreciates 13% each year.
a) What is the percent written as a decimal? ____________
b) Write an exponential decay equation.
c) How much money will the car be worth in 5 years?
d) How much money will the car be worth in 10 years?
Example 2:
Suppose a boat purchased for $28,000 depreciations by 10% each year.
a) What is the percent written as a decimal? ____________
b) Write an exponential decay equation.
c) How much money will the boat be worth in 2 years?
d) How much money will the car be worth in 5 years?
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y = the final amount
a = the initial amount
r = rate of growth/decay
t = time passed
Practice Growth and Decay
You may use a calculator to solve the problems below. Write directly on this page.
Exponential Decay
y = a(1 – r)t
Exponential Growth
y = a(1 + r)t
1) There were 220 sports radio stations in the year 2000. The number of radio stations has since
increased by approximately 14.3% per year.
a) What is the percent written as a decimal? ____________
b) Write an exponential equation.
c) What will be the value of t in the year 2015?
d) If the trend continues, how many sports radio stations will there be in 2015?
2) Marlene invested $500 in a savings account with an annual interest rate of 2.5%.
a) What is the percent written as a decimal? ____________
b) Write an exponential equation.
c) How much money will Marlene have after 20 years?
4) A new car costs $18,000. It is expected to depreciate (decrease in value) at an average rate of 12%
per year.
a) What is the percent written as a decimal? ____________
b) Write an exponential equation.
c) Find the value of the car in 8 years.
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5) The current value of Connie's car is $5300. The value of her car has been depreciating 22% per
year.
a) What is the percent written as a decimal? ____________
b) Write an exponential equation.
c) She hopes to keep the car for 4 more years. What will it be worth then?
6) In 2000, the population of Anytown was 250,000. It has been declining at a rate of 3.2%
year.
a) What is the percent written as a decimal? ____________
each
b) Write an exponential equation.
c) What will be its population in 2020?
7) In 2000, Betterville had a population of 175,000 and was growing at a rate of f 5.6% each year.
a) What is the percent written as a decimal? ____________
b) Write an exponential equation.
c) What will be its population in 2020?
8) In 2005, Sally's car was worth $2300 and has been appreciating 21% each year.
a) What is the percent written as a decimal? ____________
b) Write an exponential equation.
c) What will be its population in 2010?
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Exponent rules quick review:
Simplify the following completely
1. (10a4)(2a3)
5. (-4xy)(2x3y
2. (10a4)3
6. (-4xy)2
9.
− 4 xy
2 x −3 y
⎛ − 4 xy ⎞
10. ⎜⎜ −3 ⎟⎟
⎝ 2x y ⎠
3.
10a 4
2a 3
7.
− 4 xy
2x3 y
4.
10a 4
2a − 3
8.
2 xy
− 4x3 y
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