Math 6High ( Period 3)

Using a Least Common Denominator 3.2 

In chapter 2 we learned to find the LCM of two or more whole numbers. When you add or subtract fractions with different denominators,  a convenient denominator is the least common denominator… which is really  the Least Common Multiple of the Denominators… but it is simply called the LCD

Add:
3/8 + 5/ 12

Using the method we talked about yesterday, a common denominator is the product of 8 and 12

8(12) = 96

so we would do the following:

3(12) + 5(40)
96

36+ 40
96

76/96 

but we can simplify this. 

Using the GCF  ( OMG- we use both LCM and the GCF  to simplify fractions!!)

76/96 = 19/24

Using the LCM of 8 and 12 (now—how do we find that ?... Oh yeah.. use the BOX method or inverted division to get the prime factorization of each) So the LCM(8.12) = 24

Now rewrite the fractions

I taught you to stack them ( STACK ‘EM}

3/8 ·3/3= 9/24

5/12∙2/2 = 10/24

No you can add the numerators

9 +10
24

Wow—We arrived at the same solution!

Using the LCD reduces the amount of simplifying you need to do!

But what happens when you need to add three fractions with three different denominators? You can’t use the BOX method of finding the LCM… You need to find the LCM by… using

ALL THE FACTORS TO THEIR GREATEST POWERS… 

ALL THE FACTORS TO THEIR GREATEST POWERS!

1/3 + ¼ + 1/8  Yikes… what to do?

First try to see if one of the denominators is a multiple of the other… you now only need to find the LCD of the that fraction and the one that isn’t a multiply or factor . 

In this case,  8 is a MULTIPLY of 4 so I don’t need to worry about 4

I need to find the LCM  of 3 and 8.. Now that’s easy

LCM(3,8) is their product 24

So change all of the denominators to 24

1/3 = 8/24

¼ = 6/24

1/8 = 3/24

And add

23/24

Evaluating a Variable expression

Evaluate x – y + z  when  x = 9/10  y = 3/4 and z = 1/3

x- y + z = 9/10 – 3/4  + 1/3

Rewrite using the LCD

9/10 ·6/6 = 54/60

3/4·15/15 = 45/60

1/3·20/20 = 20/60

54 – 45 + 20
60

29/60

Algebra Honors (Periods 5 & 6)

 Factoring Pattern for x² + bx+ c, c negative 5-8

Goal- to factor quadratic trinomials whose quadratic coefficient is 1 and whose constant is negative

The method used in this lesson is  very similar to that used to factor x² +bx + c , c is  positive, except instead of the sum of the two factors you  find their difference. Remember  with x² +bx + c , c is positive, you find two numbers whose product is c and whose sum is b.

This time find two numbers whose product is c (which is negative)—so ONE of the TWO factors must be negative. You will have either  (x + )(x - )  or  (x - )(x + ) Since c is negative, one of the two factors MUST be negative.

The first sign in x² + bx + c , c is negative determines “Who wins!”  Let’s rewrite x² +bx + c , c is negative as either   x² +bx - c ,  or x² -bx - c   to see how this works.

We started with x² –x – 20
Set up your hugs…. (x -  ) (x +  )… with the winning sign going in the first set of hugs

Then using the X method find two numbers whose product is 20 and whose difference is 1 ( and in this case actually -1) We found 5 and 4 works and the 5 must be negative to get  -1

so (x -5)(x + 4)

If the quadratic was x²+x-20 you would still have the same factors 5 and 4 but this time the difference is +1 so you would have ( x +5)(x -4)

How about x² + 29a – 30

The factoring pattern is ( x + )(x -  )
Use the X method and find two numbers whose product is 30 and whose difference is 29
We find it has to be 30 and 1  (x+30)(x -1)

To check just FOIL, FireWorks, use the BOX method or just double-distribute to get back to where you started!!

x²-4kx +12k²

This time we have another variable on the last two terms so the factoring pattern starts out as
(x- _k)(x+ _k)
But again we just need to find two numbers whose product is 12 and the difference is 4
6 and 2 work so its ( x -6k)(x + 2k)

Find all the integral values for k for which the given polynomial can be factored.

c²-kc-20

For this exercise, set up a T chart with all the factors that multiply to 20

we found 1 and 20, 2 and 10, and 4 and 5.  Now taking their differences, we find that ± 19 ± 8 ± 1 all work.

Find two negative values for k  for which the given polynomial can be factored. (There are many possibilities—the class found several)

y² + 4y  + k

We found -5, -12, -77,  45, -21, … and the huge set of numbers Jeffrey King found… what were those numbers?

Math 6 High ( Period 3)

Adding & Subtracting Fractions 3.1 

To add fractions with common denominators à just add their numerators ad write this sum over the denominator.

To subtract fractions with common denominators à just subtract their numerators and write the difference over the denominator

1/7 + 3/ 7 = 4/7

7/10 – 3/1-0 = 4/10 = 2/5

In the 2^nd example we found 4/10 to be our answer but then we could simplify… Always simplify your answer.

We then drew the picture on Page 111 of the stack of books and found the height of the stack.

3/8 + 7/8 = 10/8  which is an improper fraction so we changed it to a mixed number 1 ¼  So the books were stacked  1 ¼ in

To add and subtract fractions with different denominators first rewrite the fractions so that they have the SAME denominator. Then use the rules above.

You can use ANY COMMON DENOMINATOR ( the LCD is the Best—but we will discuss that tomorrow in detail)

a/b + c/d = (ad + bc)/bd

If you cannot figure out a common denominator—one of the choices is always to multiply the denominators. That product will always be one of the common denominators!

¼ + 2/ 5 =

1(5) +4(2)
20

13/20

Math 6A (Periods 2 & 4)

Rounding 3-5 (cont'd)

Round the following number to the designated place value:
509.690285

tenths: 509.690285
You underline the place value you are rounding to and look directly to the right. If it is 0-4 you round down; if it is 5-9 you round up 1.
so here we round to
509.7

hundredths
509.690285
becomes 509.69

hundred-thousandths
509.690285
becomes
509.69029

tens
509.690285
becomes
510

(a) What is the least whole number that satisfies the following condition?

(b) What is the greatest whole number that satisfies the following condition?
A whole number rounded to the nearest ten is 520.
Well, 515, 516, 517, 58, 519, 520, 521, 522, 523, 524 all would round to 520
so

(a) 515
(b) 524

A whole number rounded to the nearest ten is 650
(a) 645
(b) 654

A whole number rounded to the nearest hundred is 1200
(a) 1150
(b) 1249
How about these...
(a) What is the least possible amount of money that satisfies the following condition?
(b) What is the greatest possible amount?

A sum of money, rounded to the nearest dollar is $57
(a) $56.50
9b) $57.49

A sum of money rounded to the nearest ten dollars $4980
(a) $4975
(b) $4984.99

Algebra Honors (Periods 5 & 6)

Factoring Pattern for x² + bx+ c, c positive 5-7

In this lesson we will be factoring trinomials that can be factored as a product of ( x +r)(x + s)
where r and s are both positive OR both negative integers.

x² + ( r + s)x + rs  

(x +3)(x+5) = x² + 8x + 15

(x – 6)(x -4) = x² -10x + 24

where -10 is the sum of -6 and -4

and

24 is the product of -6 and -4

Our book suggests that you list all the pairs of integral factors whose products equal the constant term Then, find the pair of integral factors whose SUM equals the coefficient of the linear term (remember your new vocab)

For factoring x² + bx + c, where c is positive  you only need to consider factors WITH THE SAME SIGNS as the linear term!!

In class, I showed the diamond method of calculating products and sums… and even recommended an app to practice!!
Here is the link for the iphone,  ipad app…

y² + 14y + 40

Since the linear term ( +14y) is positive you know to set up two sets of HUGS with + in the middle

(  +  )(  +  )

then you can add the single y’s since y² = y·y

(y + )(y + )

Now either list all the integral factors or use the diamond method and you discover that 10 and 4 are the two factors that multiply to 40 AND also sum to 14

(y + 10 )(y + 4)

To check if you are accurate, use FOIL or Fireworks… or the BOX method and see if you get back to the original problem!

y2 – 11y + 18

This time notice that the linear term is -11y so you will be looking for a pair of numbers whose product will be positive  ( so both need to be negative)

Set  up your  HUGS  similarly—EXCEPT both signs need to be NEGATIVE

(y -   )( y -  )

Since -11 is negative, think of the negative factors of 18 using the book’s method or the diamond method

hmmm… -9 and -2 work

(y - 9  )( y -  2)

Again To check if you are accurate, use FOIL or Fireworks… or the BOX method and see if you get back to the original problem!

A polynomial that cannot be expressed as a product of polynomials of lower degree is said to be irreducible. An irreducible polynomial with integral coefficients whose greatest monomial factor is 1 is a PRIME POLYNOMIAL.

Factor x² -10x + 14

Setting up your HUGS   you start to think what two factors multiply to 14 and SUM to 10… hmmm. NOTHING…

Therefore x² -10x + 14  cannot be factored and it is a prime polynomial

Find all the integral values of k for which the trinomial can be factored

x² + kx + 28

28 can be factored as a product—using  a T chart

list all the factors

(1)(28) (2)(14) (4)(7)

Taking the corresponding sums you get 29, 16 and 11

BUT… remember you can also have the negatives here

so the values of k can be ± 29,   ± 16,   ± 11

{-29, -16, -11, 11, 16, 29}

Math 6A ( Periods 2 & 4)

Rounding 3-5 



Our book talks about phonograph records... we needed to have a discussion about what a phonograph record is... Then we discussed what the cost of $7.98 would round to? We decided that $8 would be a good rounded number.

A general rule:
We underline the place we are rounding to and look to the number to its right. If the number is 5 or greater we round our underlined number up one and put zeros after it...
32, 567
rounded to the nearest ten thousand --> 30,000

rounded to the nearest thousand --> 33,000

rounded to the nearest hundred --> 32,600

rounded to the nearest ten --> 32,570

Rounding Decimals becomes a similar process BUT there are a few differences:

4.8637

rounded to the nearest thousandth --> 4.864

rounded to the nearest  hundredth --> 4.86

rounded to the nearest tenth  --> 4.9

rounded to the nearest unit  5

Math 6A (Periods 2 & 4)

Comparing Decimals 3-4



In order to compare decimals, we compare the digits in the place farthest to the left where the decimals have different digits.

Compare the following:


1. 0.64 and 0.68 since 4 < 8 then 0.64 < 0.68.




2. 2.58 and 2.62 since 5 < 6 then 2.58 < 2.62 .




3. 0.83 and 0.833



To make it easier to compare, first express 0.83 to the same number of decimal places as 0.833



0.83 = 0.830 Then compare



0.830 and 0.833 since 0 <3 br="br">
Then 0.830 < 0.833.



Write in order from least to greatest



4.164, 4.16, 4.163, 4.1



First, express each number to the same number of decimal places

Then compare. 4.164, 4.160, 4.163, 4.100




The order of the numbers from least to greatest is


4.1, 4.16, 4.163, 4.164

Math 6 High (Period3)

Connecting Decimals &  Fractions 2.8 

Verbal --> fraction--> decimal
3/4 to figure out what the decimal equivalence is you need to divide 3 by 4

3/4 = 0.75

But what about 5/3
5/3 = 1.666666...
It is a repeating decimal
we write
5/3 = 1.6 with a bar over the 6

The bar is called a vinculum.
Write the repeating part only once and put the vinculum ( bar) over the repeating part of the decimal.

2/7 was an interesting decimal
we found that 2/7 = 0.285714285714....
We then talked about the 1/7th family and noticed that
1/7 = .142857142857....
2/7 = .285714285714...
3/7= .428571428571...
4/7= .571428571428...
5/7= .714285714285....
6/7 = .857142857142...
all are repeating decimals with the same 6 digits in the same order... just the starting number changed...

But the textbook asked us to round 2/7 to the nearest thousands.. so we divide to the ten thousands  and found we had
0.2857 so we look to the place past the thousandths place and since it was a 7 we rounded up and dropped the digits past the thousandths place.
2/7 rounds to 0.286

Next, the textbook ased us to plot the following on a number line
2.4, 3.2, 4/5  3 2/5 and 11/5

Change all to decimals and the plot on a number line We found that 4/5 = 8/10 = 0.8
3 2/5 = 3 4/10 = 3.4 and 11/5 =  2 1/5 = 2 2/10 = 2.2
Then it became easy to plot on a number line

How do you write 2.5 as a mixed number ?  2 1/2
How about 0.125?

read it it is 125/1000  Then use divisibility rules what goes into 125 and 1000? 5 does but wait-- so does 25...
5/40  but that can be simplified because 5 goes into 5 and 40 so 5/40 = 1/8

Algebra Honors (Periods 5 & 6)

Differences of Two Squares 5-5

(a + b)(a-b) = a² - b²
(a+b) is the sum of 2 numbers
(a-b) is the difference of 2 numbers
= ( first#)² - ( 2nd #) ²

( y -7)( y + 7) = y² - 49
We did the box method to prove this.

(4s + 5t) (4s - 5t)
16s² - 25t²

(7p + 5q)(7p-5q) = 49p² - 25q²
But then we looks at
(7p+5q)(7p+5q) that isn't the difference of two squares that is
49p² + 70pq + 25q²
So let's look at the difference of TWO Squares:

b² -36
So that is ( b + 6)(b -6)

m² - 25
(m + 5)(m -5)

64u² - 25v²
(8u + 5v)(8u -5v)

1 - 16a²

(1+4a)(1- 4a)

But what about 1- 16a⁴

( 1 + 4a²)(1 - 4a²) but we are NOT finished factoring because
(1 - 4a²) is still a difference of two squares so it becomes

( 1 + 4a²)(1 + 2a)(1 - 2a)

t⁵ - 20t³ + 64t

Factor out the GCF first

t(t⁴ - 20t² + 64)
t(t² -16)(t² -4)
YIKES... we have two Difference of Two Squares here...

t(t +4)(t-4)(t +2)(t -2)



81n² - 121
(9n +11)(9n -11)
3n⁵ - 48 n³


Factor out the GCF

3n³ (n² - 16)
3n³(n +4)(n-4)

50r⁸ - 32 r²

Factor out the GCF
2r²(25r⁶ - 16)

2r²(5r³ +4)(5³ -4)

u² - ( u -5) ²
think a² - b ² = (a + b)(a -b)

so
u² - ( u -5) ² =
[u + (u-5)][u - (u-5)]
(2u -5)(5)
= 5(2u-5)

t² - (t-1)²

[t +( t+1)]{t-(t-1)]
2t-1(+1)
=2t -1


What about x²ⁿ - y ⁶ where n is a positive integer

well that really equals
(xⁿ)² - (y³)²
so
(xⁿ + y³)(xⁿ - y³)

x²ⁿ - 25
(xⁿ + 5)(xⁿ - 5)

a⁴ⁿ - 81b⁴ⁿ
(a²ⁿ + 9b²ⁿ)(a²ⁿ - 9b²ⁿ)
= (a²ⁿ + 9b²ⁿ)(aⁿ + 3bⁿ)(aⁿ - 3bⁿ)


When multiplying to numbers such as (57)(63)
think
(60-3)(60 +3)
then the problem becomes so much easier

3600 - 9 = 3591 DONE!!!

(53)(47) = (50 +3)( 50-3)
2500 - 9 = 2491

Math 6A (Periods 2 & 4)

Decimals 3-3

Although decimals ( termed decimal fractions) had been used for centuries, Simon Stevin in the 16th century began using them on a daily basis and he helped establish their use in the fields of sciences and engineering.

Note that
1/10 = 1/10¹
1/100 = 1/10²
1/1000 = 1/10³

We also know that
1/10= 0/1
1/100 = 0.01
1/1000 = 0.001
1/10000 = 0.0001
and so on... these strings of digits are called decimals.


SO 1/10 = 1/10¹= 0.01 and it is equal to 10^-1
Notice that 10^-1 is NOT a negative number-- it is a small number
and 10^-21 is not a negative number it is a VERY TINY number

AS with whole numbers, decimals use place values. These place values are to the RIGHT of the decimal point.
We need to be able to write decimals in words as well as expanded notation.
In class we used 0.6394 as our example

zero and six thousand three hundred ninety-four ten-thousandths.

Notice how this number when written in words begins...with "ZERO AND"
Why do we need to do that?

Also notice that there is a hyphen between ten and thousandths in ten-thousandths. It is critical to understand when you must place a hyphen.
We read the entire number to the right of the decimal point as if it represented a whole number, and then we give the place value of the digit farthest to the right.

So, although 0.400 is equivalent to 0.4
we must read 0.400 as "zero and four hundred thousandths."

Now look at the following words
"zero and four hundred-thousandths." What is the subtle difference between those two phrases above?
There is a hyphen in the last phrase-- which means that the hundred and the thousandths are attached and represent a place value so

zero and four hundred-thousandths is 0.00004 while
zero and four hundred thousandths is 0.400

Carefully see the distinction!!

Getting back to our 0.6394

to write it in decimals sums and then in exponents:
0 + 0.6 + 0.03 + 0.009 + 0.0004

0 + 6(0.1) + 3(0.01) +9(0.001) + 4(0.0001)

0(10⁰) + 6(10^-1)+ 3(10^-2)+ 9(10^-3)+ 4(10^-4)

14.35 is read as fourteen AND thirty-five hundredths.
When reading numbers, only use the AND to indicate the decimal point

Algebra Honors (Periods 5 & 6)

Multiplying Binomials Mental 5-4

Look at
(3x - 4)(2x+5)

remember FOIL
First terms (3x)(2x)
Outer terms (3x)(5)
Inner Terms (-4)(2x)
Last Terms (-4)(5)

6x² + 15x -8x -20
6x² + 7x - 20
Or use the box method as we have done in class
This is a quadratic polynomial
The quadratic term is a term of degree two

Remember a linear term has a term of degree 1 such as y = 3x + 5

6x² + 7x - 20

The 6x² is the quadratic term
the +7x is the linear term
and the - 20 is the constant term

(x +1)(x +3) = x² + 4x + 3

(y + 2)( y + 5) = y² + 7y + 10
( t -2)( t -3) = t² -5t + 6

( u -4)(u -1) = u² - 5u + 4

What about
( u-4)(u +1) = u ² -3u -4

See the difference between the two?

(7 - k)(4 -k)

28 - 11k + k²


r + 3)(5 - 5)
r² - 25 - 15



(3x - 5y)(4x + y)
12x² - 17xy - 5y²

a + 2b)(a-b)

careful....
a² + ab - 2b²

n(n-3)(2n+1)
first distribute the n
(n² -3n)(2n +1)
2n³ - 5n² - 3n

Solve for
(x-4)(x +9) = (x +5)(x -3)
x² + 5x - 36 = x² + 2x -15
5x - 36 = 2x - 15
3x = 21

x = 7
or in solution set notation {7}

Math 6High (Period 3)

Mixed Numbers & Improper  Fractions 2.7 

What are Mixed Numbers?  --> sum of a whole number and a fraction
6 3/5 = 6 + 3/5

improper fraction -->  numerator is greater than or equal to the denominator.
4/3 and 8/8 are both improper

Write 2 2/5 as an improper faction
2 + 2/5  10/5 + 2/5 = 12/5

3 1/4 = 13/4

2 5/6 = 17/6


Write an improper fraction as a mixed number

7/3 =  2 1/3

5/4 = 1 1/4

11/6 = 1  5/6

16/5 = 3  1/5

order the following from least to greatest

4/3,  5/6,   1,  2/3,   1 1/6

first notice two are smaller than 1 and two are larger
Then all you have to do is compare the two smaller than 1
2/3 < 5/6

compare the two that are larger than 1

1 1/6 < 4/3
so
2/3 < 5/6 < 1 < 1 1/6 < 4/3

Refer to your notes on the various ways to compare two fractions.

Then we ordered numbers on a number line.
We graphed 7/5 on a number line  changing 7/5 to 14/10 to make it easier to plot.

Math 6A ( Periods 2 & 4)

The Decimal System 3-2

Our system of numbers uses the following ten digits:
0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9

Whole numbers greater than 9 can actually be represented as sums. For example
386 = 300 + 80 + 6
or
3(100) + 8(10) + 6(1)

Notice that each place value is ten times the value of the place value to its RIGHT!!
The number 10 is called the BASE of this system of writing numbers.
The system itself is called the DECIMAL SYSTEM from the Latin word decem-- which means ten
Think December- but why is that month the 12th month? hmmm.. Did anyone know from class?

Look at the chart given to you in class and notice the place names for the first several numbers.
To make numbers with MORE THAN four digits easier to read, commas are used to separate the digits into groups of three-- starting from the RIGHT
Think of the comma as a 'signal' for you to mention a place name category!!

In words the number 420,346 is written as
"four hundred twenty thousand, three hundred forty-six."

The expanded notation for 420,346 is given by

4(100,000) + 2(10,00,000) + 0(1000) + 3(100) + 4(10) + 6(1)

Using exponents the expanded notation may be given as
4(10⁵) + 2(10⁴)+0(10³)+3(10²)+4(10¹)+6(10⁰)

What hmmm.. how is (10⁰) = 1
any number to the zero power is 1.


From yesterday we noticed that
10² ⋅ 10³ = 10⁵
and
10⁶ ⋅ 10³ =10⁹
so we wrote a rule for any exponent values a and b
10^a ⋅ 10^b = 10^a+b
Then I asked what would be a rule for

10⁸
10⁵
and
10⁶
10⁴
Students decided that since we added the exponents when multiplying we could subtract the exponents when dividing!!

so could we write a rule for any exponent values a and b?
YES
10^a
10^b
= 10^a-b
but then
10⁵
10⁵
would be
10 ^5-5 which is 10⁰
BUT WAIT... isn't any number divided by itself equal to ONE. That is,
10⁵
10⁵
= 1
So
10⁰ = 1

279,043
is read as two hundred seventy-nine thousand, forty-three.
and in expanded notation becomes:
2(100,000) + 7(10,000) + 9(1000) + 0(100) + 4(10) + 3(1)

or with exponents:
2(10⁵) + 7(10⁴) + 9(10³) + 0(10²) + 4(10¹) + 3(10⁰)
Writing a variable expression to represent two or three digit numbers requires you to think of the value of each place.
For example,
The ten's digit is t and the ones' digit is 2
you can't just put t + 2 WHY???

Let's say you are thinking of the number 12 when we said the expression
"The ten's digit is t and the ones' digit is 2"
If you said t + 2 you would get 1+2
and that = 3
It isn't the two digit number we wanted--- 12.

so what is the place value of the 1?
It is really in the ten's place or written as 1(10)
To write the variable expression we must in include the value

10t + 2 becomes the correct expression

What about the ten's digit is 5: the ones' digit is x? 5t + x. Do I need to put a 1 in front of the x for the ones' digit? No it is... invisible!!

Some of you asked that I put the Magic Number Trick that we did online... try this one on your parents...

Have your 'victim' follow these directions:
Choose any four-digit number
Write the thousand's digit
Write the thousands' and hundreds' digits
Write the thousands' hundred's and tens' digits.
Add these numbers

Multiply by 9

Find the sum of the digits of the original number
add this sum to the previous results.

Your 'victim's' answers should always be the same as their original number chosen.
Example
I used in class 1492
Write the thousand's digit-------> 1
Write the thousands' and hundreds' digits ---> 14
Write the thousands' hundred's and tens' digits--->149
Add these numbers 1 + 14 + 149 = 164
Multiply by 9 or 164(9)= 1476

Find the sum of the digits of the original number 1 + 4 + 9 + 2 = 16
add this sum to the previous results. 1476 + 16 = 1492

Voila!! It worked!!
Email me your parent's comments or post them here!!

Math 6A (Periods 2 & 4)

Exponents and Powers of Ten 3-1
When two or more numbers are multiplied together--each of the numbers is called a factor of the product.

A product in which each factor is the SAME is called a power of that factor.

2 X 2 X 2 X 2 = 16. 16 is called the fourth power of 2 and we can write this as
2⁴ = 16

The small numeral (in this case the 4) is called the exponent and represents the number of times 2 is a factor of 16.
The number two, in this case, is called the base.

When you are asked to evaluate... simplify... solve... find the answer
That is,
Evaluate
4³ = 4 X 4 X 4 = 16 X 4 = 64

The second and third powers of a numeral have special names.
The second power is called the square of the number and the third power is called the cube.

We read 12² as "twelve squared" and to evaluate it
12² = 12 X 12 = 144

Powers of TEN are important in our number system.
Make sure to check out the blue sheet and glue it into your spiral notebook
First Power: 10¹ but the exponent is invisible = 10
Second Power: 10² = 10 X 10 = 100
Third Power 10³ = 10 X 10 X 10 = 1000
Fourth Power 10⁴ =10 X 10 X 10 X 10 = 10,000
Fifth Power 10⁵ = 10 X 10 X 10 X 10 X 10 = 100,000

Take a look at this list carefully and you will probably see a pattern that we can turn into a general rule:

The exponent in a POWER of TEN is the same as the number of ZEROS when the number is written out.

The number of ZEROS in the product of POWERS OF TEN is the sum of the numbers of ZEROS in the factors.

For example Multiply.
100 X 1000
Since there are 2 Zeros in 100 and 3 zeros in 1000,
the product will have 2 + 3 , or 5 zeroes.
100 X 1000 = 100,000

When you need to multiply other bases:

first multiply each
For example

3⁴ X 2 ³ would be
(3 X 3 X 3X 3) X ( 2 X 2 X 2)
= 81 X 8 = 648

What happens when you multiply the same bases?
3⁴ ⋅ 3² = 3⋅3⋅3⋅3⋅3⋅3 or 3 ⁶
We just add the exponents if the bases are the same!!

Well then, what about (3⁴)² ?
Wait.. look carefully isn't that saying 3⁴ Squared?
That would be (3⁴)(3⁴), right?
.. and looking at the rule above all we have to do here is then add those bases or 4 + 4 = 8 so the answer would be 3⁸.
OR
we could have made each (3⁴) = (3⋅3⋅3⋅3)
so (3⁴)² would be 3⋅3⋅3⋅3⋅3⋅3⋅3⋅3 or still 3⁸
But wait... isn't that multiplying the two powers? So when raising a power to a power-- you multiply!!
(3⁴)² = 3⁸

1 to any more is still just 1
1⁵ = 1

0 to any power is still 0!!

Evaluate if a = 3 and b = 5
Just substitute in... but use hugs () we all love our hugs!!
a³ + b²
would be (3)³ + (5) ²
= 27 + 25 = 52




Check out this great Video on the Powers of Ten
POWERS OF TEN