Introduction to Exponents

Introduction to Exponents - Transcript

2 6 1 Introduction to Exponents
Repeated Multiplication
Remember multiplication is a way to write repeated addition To say 3 3 3 3 we write 3x4
Sometimes multiplication is done over and over and over To write 3x3x3x3 we write 34
42 read four squared means 4x4 so 42 16 53 read five cubed is 5x5x5 125
Practice Write the indicated operation using exponents Then perform the operation
a 4x4x4
10X10X10X10X10X10 2x2x2x2x2x2x2x2
b
5x5 3x3x3x3x3 9x9x9
c
0 3 x 0 3 x 0 3 5 1 x 5 1 1 2 1 2 1 2
d 0 2 x 0 2 x 0 2 x 0 2
0 01x0 01x0 01x0 01 1 5x1 5
e 3 x 3 x 3 x 3
100x 100 x 100 1 x 1 x 1 x 1 x 1
f aaaaa answer a5
xxxxxxxxxxx 2f2f2f2f

Practice Perform the indicated operation
g
25 72 82
h
122 152 252
i
4 2 10 2 13 2
j
4 3 12 2 5 3
k
10 3 0 5 4 0 2 4
Review Topic 7 Fraction Multiplication and Division
Look at the following examples Decide what the rule is for multiplying fractions
4 5 x 3 4 12 20 3 5
Notice Fraction numbers must be written as the ratio of two integers to be multiplied
Thus to multiply 2 x 1 x 12
First change the mixed numbers to improper fractions 5 2 x 7 4 x12 1
to get 420 8
This can be simplified to 52
One other step can save time Canceling reduces the answer before you multiply the numbers

Fraction Division
To divide fractions invert the second fraction and multiply
2 3 2 3 x 2 1 4 3 1 1 3
Notice DON T cancel until after the divisor second number is inverted
For word problems be very careful which number to write first in the problem The thing being split cut apart or sorted goes first

Change to improper
Invert and Multiply
Cancel
Multiply
Simplify
Practice






f 53
3 4 2 7 8 2
g Jack has 2 sacks of flour If he uses 2 3 of the flour how much flour is left

Tiffany splits 4 5 pounds of nails into 4 piles How many pounds in each pile
h 15 pounds of chocolate is put in pound boxes How many full boxes are made How much chocolate is left
A cube box with all sides the same length is 3 inches on each side What is the volume of the cube
Hint Multiply all three sides 3 3

i 5 purdinkles are split into buckets that hold 2 3 of a purdinkle How many buckets will we need
You don t need to know what a purdinkle is

Find the surface area of the cube above
Hint 6 times 3 2

j 2 yards of fabric are needed for each shirt How many yards for 10 shirts
How many shirts could be made from 30 yards
How many cubes can be made of a piece of metal 367 square inches
How much metal is left over

k Write a word problem that requires multiplication of fractions then subtraction to solve
2 6 2 Word Problems

a If one yard of polar fleece fabric costs 12 00 how much does 5 yards cost

One cubic foot of water weighs 62 pounds How much does 5 3 8 cubic feet of water weigh
b Janet worked with students for 5 hour on Tuesday 4 3 8 of an hours on Wednesday and 1 2 3 hours Thursday How much time did she spend with students for the three days

My mileage gauge reads 4533 3 10 miles
Ok I admit it has rolled over
Last week it read 4042 7 10 miles
How far have I driven this week

c The shopping list included 3 pounds of hamburger 3 pounds of apples 2 pounds of oranges and 1 3 pound kiwi How many pounds of fruit were purchased

The baby was 6 2 5 pounds at birth
two months ago
He gained 4 pounds
How much does he weigh now
d How many pieces of wire each 3 inches long can be cut from a roll that is 50 inches long

One slide is 5 3 8 feet taller than the other
The short slide is 12 feet
How tall is the tall slide
e The records showed 123 pounds of monkey food were still in the inventory Each day 25 2 3 pounds are used How much is left after three days

Monster costumes for the play each required 5 3 5 yards of fabric How many yards will be needed to make all 12 costumes
f How many quart jars of jam can be made from 34 5 8 pound of strawberries 2 3 of a pound will make one quart jar

How many 1 2 3 pound boxes of cookies can be made from 10 pounds of cookies
g Jamie spent 3 hours typing word problems 2 hours making answer keys 2 3 of an hour making copies One twelfth of her time was spent yelling at the computer The rest of her eight hour day was spent filing How much time was spent filing

One section of a wall of my house will be a book shelf The section is 6 feet long and 10 feet high Most shelves should be 1 foot tall but the two bottom shelves will be 1 feet tall for oversized books This will leave the top shelf an odd height How much shelving is needed Don t put shelving on the ceiling but the floor will need shelving

h While doing home health visits Jack drove 3 miles to Mrs Jenkins 5 7 10 miles to the next appointment 4 2 5 to
Mr Smith and 7 miles back to the office What should he claim on his mileage report

Jon will work overtime for time and a half He works a 10 hour day His regular pay is 12 50 per hour How much did he make
i The family got three 2 pound boxes of chocolate for Christmas How much chocolate does each of the five people get

Larry was home from work because he was ill for 5 12 of his 40 hour work week How many hours did he work
j Jon s full truck weighs 3 tons The average crate weight 2 5 ton If there are seven crates on the truck how much does the empty truck weigh

If you change the rotating speed of a cement mixer from 12 rpm s to 5 1 4 rpm s how many revolutions per minute slower does the cement mixer spin
k Jackie lost 10 3 5 lbs Before the diet she weighed 140 lbs How much does she weigh now

Rosy cut 3 feet from a 10 foot piece of sheet metal How long was the remaining piece
l Zuella made some dresses out of 15 yards of fabric Each required 3 2 3 yards How many dresses was she able to make

I only have pounds of German baking chocolate left My recipe calls for 5 8 pounds How much will I have left
m A ten pound bag of cotton balls has been opened It s a really big bag Yesterday there was 8 2 3 pounds left If someone uses 7 8 of a pound how much do the remaining cotton balls weigh

Todd worked 9 hours Monday 10 hours on Tuesday 8 hours on Wednesday 8 hours on Thursday and 9 hours on Friday He makes 14 per hour regular pay and time and a half for anything over 40 hours How much did he make that week
n Craft trim is to be cut in 5 3 8 in pieces How many pieces can be cut from 3 yards

The trip to the store is 7 8 miles Joan walked 2 3 of the way How far has she walked

2 6 3 Exponent Rules




don t worry what fractional exponents mean yet

Practice Simplify






When dividing like bases subtract the exponents

Practice Simplify



This leads to another rule

Practice Write each answer two ways as a fraction with positive exponents and without using fractions but using negative exponents











or all cancel for an answer of 1
Anything to the zero power is 1
Practice Simplify
o 30 x0 4y 0 3st5 0 5 8 9 7 12 0
p 4x0 8e0 2s5x0 2 3st5 0 3st5 0

using a previous rule
An easier way would be to multiply the 3 and 5 to get the new exponent of 15
Practice Simplify





are not like terms and cannot be combined
The multiplication can be done
Do add the exponents


c 4s 7s 8s2
4s 7s 8s2 7r4 7r3 7r4 7r3
d 3x2 4x2 8x2
3x2 4x2 8x2 w 6w 7w2 w 6w 7w2

Mixed practice Simplify







2 6 4 Scientific Notation
Scientific notation is used to write very large and very small numbers
The distance to the sun is150 000 000km or 1 5x108km
The bacteria Streptococcus pyogenes is 0 0000625 cm or 6 25x10 5cm long
4x103 4x10x10x10 4x1000 4000 Notice the exponent on the 10 is 3 and the decimal after the 4 moved to the right 3 spaces
2 3x105 23000 moving the decimal right 5 spaces
2 3x105 is scientific notation and 23000 is decimal notation
Practice Write the following in decimal notation
a 4 6x108 5 2x103
3 4x1012
b 7x103 8X108
2 3x108 23x107 0 23x109 230x106 But only one of these is in scientific notation
2 3 is between 1 and 10 so the correct form of the number in scientific notation is 2 3x108
Although the other numbers are the same as 2 3x108 they are not scientific notation
3 21x102 is the scientific form of 321
To write 45600000 in scientific notation 4 56x107 45600000
move the decimal so only one digit is in front and drop ending zeros 4 56
Multiply by ten to the power that matches the number of spaces the decimal was moved
Practice Write the following in scientific notation
a 8 900 000 000 000
2 000 000 000 000 000 178 000 000 000 000
b 1 000 000 000 000
24 000 000 000 000 145 000 000 000 000 000
c 8 431 000 000 000 000 1 000 000 000 000 000 000 000 000 5 000 000 000 000

0 00068 Dividing by 10000 moves the decimal 4 spaces to the left
To change a small number in scientific notation to decimal notation move the decimal to the left the number of spaces indicated by the power of 10
5 43x10 3 0 00543 left and 6 43x103 6430 right
Practice Write the following in decimal notation Watch the signs
a 4 6x10 8 5 2x10 3
3 4x10 12
b 7x10 3 1 8X10 8
1x10 10
c 1 123x10 9
5 453x10 5 6 5874x10 3
d 4 8x103
2 3x10 5 4 5x10 8
e 3x10 5 3 3x10 6 8 1x10 9

To write a small decimal notation number to scientific notation move the decimal after the first nonzero digit Count how many spaces the decimal was moved This is the power of 10 the number would be divided by Make it a negative exponent

Practice Write the following in scientific notation Watch the signs
a 0 000000043 0 00000254 0 000023
b 0 0000235 0 00007 0 0000000001
c 0 000000000028 0 00000000000009 0 00001

Practice Write the missing numbers
Decimal Notation Scientific Notation
d
The population of the United States is 3x108

e The population of the world is about 6230 000 000
f
The human heart is 3 1x100 inches in diameter

g The diameter of an artery is 00621 inches

h
The moon is 3 8x105 km away
i The mass of the moon is 73 500 000 000 000 000 000 000

j
The mass of the Earth is 5 97x1024 kg
k
A hair grows 2 08x10 4 mm per minute

Multiplication and Scientific Notation
To multiply 3 1 x 104 X 2 3x105 use the commutative property 3 1x2 3x104x105 and the associative property grouping of multiplication doesn t matter to get 7 13x104x105
Then use the exponent rules to get the final result 7 13x109
3x108 1 5x1023 4 5x1031 2 1x104 4x10 12 8 4x10 8 4 6x108 5 8x106 26 68x1014
Notice What is wrong with the last example Although the answer is correct the number is not in scientific notation To finish the problem move the decimal one space left and increase the exponent by one
26 68x1014 2 668x1015
0 00042x109 4 2x105
7890x103 7 89x106
Practice Multiply then write the following in scientific notation





f The population of the world is 6 23x109 and the average person consumes 5x103 grams of sugar per year How many grams of sugar are consumed in the world per year

Division and Scientific Notation

Notice the decimal and exponent

Practice Divide the following then write the answer in scientific notation


c A certain string of bacteria is 3 4x10 5 cm long There are approximately 80 cells in a string How wide is each single bacteria

Calculator notation
Some calculators show scientific notation the same way we have written them here
Some use EE They write 3 456 EE5 to mean 3 456 x105
Some write the same thing with this in the screen 3 45605
There are several ways electronic equipment displays scientific notation Look in your user s manual to see how it is displayed on your calculator
2 6 5 Roots
because 5 times 5 is 25
When working a square root problem ask
What times itself is the number inside the root symbol
because 3x3x3 is 27 The small 3 outside the root symbol tells how many times the answer must be multiplied to get the number inside the root
because 2x2x2x2x2 or 25 32
Practice Simplify


Prime factorization is writing a number using multiplication of only prime numbers
12 can be written as 3x4 but 4 is not prime and can be written as 2x2
so the prime factorization of 12 3x2x2 This can be written 3x22
To write 330 using its prime factorization start breaking it up into smaller factors until there are no more composite numbers
330 33x10 3x11x2x5
Practice Write the prime factorization for the following
a
12 8 27 100
b
56 28 36 32
c
50 42 18 25
d
75
48 360 72
e
7875
50176 7200 3136
f
405
432 5488 9375
g
864
442368 3600 1225

Simplifying roots You won t be using the root button on your calculator for these
First write the number as it s prime factorization
Then because this was a square root a pair of 3s can be simplified to a 3 outside the root To get an exact answer leave the 5 inside the root rather than using a calculator
four 2 s and four 3 s and fourth root
This is the simplest form of the root
Practice Write the roots in simplest form







Using a calculator will give an approximation
11 81 rounded to the nearest hundredth
To use your calculator you need to learn another notation
On your calculator find the xy button or yx Type 52 then hit the yx button Then type 25 2 69 Rounded
Practice First rewrite the root using a fractional exponent Then use your calculator to find the answer rounded to the nearest hundredth






All the rules that work for exponents work for these fractional exponents This is learned and practiced in future math classes
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and m is an integer
Prime number a number with only factor of 1 and itself
2 3 5 7 11 13 17
19 23 are prime numbers
15 is not prime because 3 and 5 also divide it evenly
15 is a composite number