Algebra Honors (Period 6 & 7)

Powers of Monomials 4-4


POWER TO ANOTHER POWER


MULTIPLY the POWERS
(m⁵)³ = m¹⁵


To check, EXPAND it out:
(m⁵)(m⁵)(m⁵) = m¹⁵



PRODUCT TO A POWER

DISTRIBUTE the power to EACH FACTOR
(m⁵n⁴)³ = m¹⁵n¹²



(a^m)ⁿ = a ^mn
(u⁴)⁵ = u²⁰
(2x)³ = (2x)(2x)(2x) = 8x³

(ab)^m = (ab)(ab)(ab).... -->m factors<--- = (a⋅a⋅a⋅a⋅a ...)(b⋅b⋅b⋅b⋅b ...) where a is multiplied m number of times and b is multiplied m number of times.... (ab)^m = a^mb^m
To find the power of a product, you find the power of each factor and then multiply

Simplify
(-2k)⁵
= (-2)⁵ k⁵ = -32k⁵

Evaluate if t = 2
a) 3t³
b) (3t)³
c) 3³t³
d) -(3t)³


Simplify
(-3x²y⁵)³
(-3)³(x²)³(y⁵)³
-27x⁶y¹⁵



RAISING A QUOTIENT TO A POWER:


DISTRIBUTE THE POWER to the numerator and the denominator

(m²/n⁶)³ = m⁶/n¹⁸

1/n⁶ = n^-6
so what does
1/m^-7 =?

Let's read it in math terms
it is 1 divided by 1/m⁷ .. and what do you do when you need to divide by a fraction? You multiply by its reciprocal so

1 divided by 1/m⁷ = 1 ÷ 1/m⁷ = 1× m⁷/1 = m⁷

Algebra Honors (Period 6 & 7)

Multiplying Monomials 4-3
POWER RULES:


MULTIPLYING Powers with LIKE BASES:

Simply ADD THE POWERS

m⁵m³ = m⁸

You can check this by EXPANDING:
(mmmmm)(mmm) = m⁸



DIVIDING Powers with LIKE BASES:

Simply SUBTRACT the POWERS

m⁸/m⁵ = m³     


Again, you can check this by EXPANDING:
mmmmmmmm/mmmmm = mmm

ZERO POWERS:

Anything to the zero power = 1


(except zero to the zero power is undefined)


Proof of this was given in class:

1 = mmmmmmmm/mmmmmmmm
= m⁸/m⁸
= m⁰ (by power rules for division)
       


By the transitive property of equality : 1 = m⁰


Review the odd/even rule

IF THERE IS A NEGATIVE INSIDE PARENTHESES:

Odd number of negative signs or odd power = negative

Even number of negative signs or even power = positive


EXAMPLES:
(-2)⁵ = -32

(-2)⁴ = +16



IF THERE IS A NEGATIVE BUT NO PARENTHESES:

ALWAYS NEGATIVE!!!!

-2⁵ = -32

-2⁴ = -16

JUST REMEMBER
NEGATIVE POWERS MEANS THE NUMBERS ARE FRACTIONS


They're in the wrong place in the fraction

m³/m⁵ = m^-2
        

m³/m⁵ = mmm/ mmmmm
= 1/mm


Again, by transitive property of equality:

m³/m⁵ = m^-2 = 1/m²

Remember the rule of powers with (  ) 
When there is a product inside the (  ), then everything inside is to the power!

If there are no (  ), then only the variable/number right next to the power is raised to that power.

3x^-2 does not equal (3x)^-2
The first is 3/x² and the second is 1/9x²

RESTATE A FRACTION INTO A NEGATIVE POWER:

1) Restate the denominator into a power

2) Move to the numerator by turning the power negative


EXAMPLE: 
1/32
 = 1/(2)⁵ = (2)^-5

Algebra Honors (Period 6 & 7)

Adding & Subtracting Polynomials 4-2

Terms to Know:
Polynomials = SUM of monomials

Monomials must have variables with whole number powers.
no variables in the denominator, no roots of numbers!!
so 1/x is not a monomial
neither is x ^1/2

constants have whole number power of zero..
7 is really 7x⁰
constants are also called constant monomials


1 term = monomial
2 terms = binomial
3 terms = trinomial

TERMS are separated by addition
( if see subtraction-- THINK: add the opposite!!)

Coefficient - number attached to the variable ( it can be a fraction)
3x² - 10x
the coefficients are 3 and -10. Make sure to attach the negative sign to the coefficient -- and ADD the OPPOSITE

y/6 is really (1/6)y so the coefficient is 1/6
if you have -x/3 that is really (-1/3)x so the coefficient is -1/3


Constant = the number that is not attached to ANY variable

Two monomials that are exactly alike ( except for the coefficient) are said to be similar, or like, terms.
-5xy² and xy² and (1/3) xy² are like terms. So is 16yxy because when you combine it 16yxy becomes 16xy²
But... -3xy² and -3x²y are NOT!!!

A polynomial is simplified when no two terms are similar
-6x³ + 3 x² + x² + 6x³ - 5 can be simplified to
4x² - 5



Some MORE TERMS YOU NEED TO KNOW
Degree of a variable in a term = number of times that variable occurs as a factor.
Degree of a term = SUM of the exponents of all its variables
-6x⁴ : the degree is 4
8x² : the degree is 2
-2x : the degree is 1
9 : the degree is 0 ( think 9 is really 9x⁰

Degree of a polynomial - HIGHEST degree of any of its terms
so
-6x⁴ + 8x² + -2x + 9
The degree of the polynomial is : 4

Leading term = term with the HIGHEST degree
Leading coefficient- the coefficient of the leading term






Descending order- write the variables with the highest power first ( This is the way it is usually written)

Ascending order- write the variables with the lowest power first ( actually NEVER used in practice)


Adding Polynomials
This is nothing more than combining LIKE TERMS
LIKE TERMS = same variable AND same power

You can either do this using 3 different strategies:
1. Simply do it in your head, but keep track by crossing out the terms as you use them.
2. Rewrite putting the like terms together (commutative and associative property)
3. Rewrite in COLUMN form, putting like terms on top of each other like you do when adding a column of numbers.

EXAMPLE OF COLUMN FORM:
(5x⁴ - 3x² - (-4x) + 3) + (-10x⁴ + 3x³- 3x² - x + 3)
Rewrite in column form, lining up like terms:


Subtraction of Polynomials
You can use the ADDITIVE INVERSE PROPERTY with polynomials!
Subtracting is simply adding the opposite so.............
DISTRIBUTE THE NEGATIVE SIGN TO EACH TERM!!
(Change all the signs of the second polynomial!)
After you change all the signs, use one of your ADDING POLYNOMIAL strategies!
(see the 3 strategies listed above under Chapter 5-7)

EXAMPLE OF COLUMN FORM:
(5x⁴ - 3x² - (-4x) + 3) - (-10x⁴ + 3x³- 3x² - x + 3)
Rewrite in column form, lining up like terms:
5x⁴ - 3x² - (-4x) + 3
- ( -10x⁴ + 3x³- 3x² - x + 3)
-----------------------------------

For the sake of showing you here, I have added ZERO Terms to line up columns
+ 5x⁴ + 0x³ - 3x² -(-4x) + 3
-(-10x⁴ +3x³- 3x² - x + 3)
-----------------------------------

DISTRIBUTE THE NEGATIVE, THEN ADD:
5x⁴ + 0x³ - 3x² - (-4x) + 3
+10x⁴ -3x³ +3x² + x - 3
-----------------------------------
15x⁴ - 3x³ + 5 x

Algebra Honors (Period 6 & 7)

Exponents 4-1
In the expression 5⁴ the number 4 is called the exponent and the number 5 is called the base
We call 5⁴ the exponential form of 5⋅5⋅5⋅5 (which is the expanded form)
The exponent tells you the number of times the base is used as a factor.
bⁿ = b⋅b⋅b⋅b⋅b⋅⋅⋅⋅b ( a total of n factors)
The expression bⁿ tells you that b is used as a factor n times.

-2⋅p⋅q⋅3⋅p⋅q⋅p written in exponential form is -6p³q²

Be careful when an expression contains both parentheses and exponents
(2y)³ = (2y)(2y)(2y) = 8y³
and
2y³ means 2⋅y⋅y⋅y = 2y³
-3⁴ = -81
(-3)⁴ = (-3)(-3)(-3)(-3) = +81
(1 + 5)² = 36
1 + 5²= 26
Simplify
(x –y)³/2x + y
for x = 2 and y = 5

-3

Simplify
x⁴ – a⁴
for x = 2 and a = 3
-65

[2³ + 3³] ÷ [2³ + (-1)²] = (8 + 27) /9 = 35/9


Evaluating terms - this is what we have been doing all year... plug it in, plug it in!!
Remember to ALWAYS put the number you substitute in parentheses!!

2x²y + 5xy - 4, where x = -4 and y = 5
Substitute carefully:

2(-4)²(5) + 5(-4)(5) - 4
= 2(16)(5) +(-20)(5) - 4
= 160 +(-100) -4
= 160 -104
= 56

Math 6 Honors ( Periods 1, 2, & 3)

Writing Inequalities 2-3

2 < 7 and 7 > 2 are two inequalities that state the relationship between the numbers 2 and 7

2 < 7 reads 2 is the less than 7 and 7 > 2 reads 7 is greater than 2
The symbols < and > are called inequality symbols.

Notice the mathematical sentence (inequality)
Two is less than ten or 2 < 10 is different from the mathematical phrase (expression) Two less than ten. 10 - 2 A number 2 + x is greater than a number t 2 + x > t

The point of the number line that is paired with a number is called the graph of that number.

Check out the graph in the middle of page 39 of our textbook. When you graph numbers on the number line, make sure to place a dot DIRECTLY ON the number line at that particular number's location.
Again, check out our textbook for examples!!

Looking at the graph of numbers, we see that the larger number will be to the right of the smaller number.

A number n is between 6 and 12 so 6 < n < 12 or 12 > n > 6

Thursday's Lesson: continuing on ....

Notice the subtle differences in the sentence
Six is greater than a number t
and the phrase
six greater than a number t

Six is greater than a number t becomes 6 > t
while
six greater than a number t becomes t + 6

What about the following inequality:
A number p is greater than a number q
is p > q

The value in cents of d dimes is less than the value in cents of n nickels.

If you need to-- set up your T-charts (refer to your class notes) one for dimes and the other for nickels.

10d represents the number of dimes and 5n represents the number of nickels

so 10d < 5n

Math 6 Honors (Period 6 and 7)

Writing Mathematical Equations 2-2

The process of writing equations really is just writing two equal expressions and joining them by an equals sign. The words "is" "equals" or "equal to" all indicate that two phrases NAME the same number.
The equals sign is the VERB in a mathematical sentence-- without it you have a mathematical expression!!
Eight increased by a number x is equal to thirty-seven.
In translating this mathematical sentence, I always start but placing the equals sign directly under the words "is equal"
so my first step would be

Eight increased by a number x is equal to thirty-seven.
---------------------------------> = <------------------
Then I would translate each mathematical phrase separately.
Yes, thirty-seven is a mathematical phrase!!
8 + x = 37

Ten is two less than a number n
10 = n - 2

Twice a number w equals the sum of the number and four
2w = n + 4

Notice that when you are indicating multiplication the number (coefficient) ALWAYS is placed in front of the variable.

So three times a number b would be 3b
The only time you see the letter first-- is when you are looking for our ROOM--
which is K 101... a mathematician did not label the room numbers!! :)

Sometimes we need to write an equation for a word sentence that involves measurements. MAKE SURE that each side of the equation uses the SAME UNIT of Measurement!!
For example. Write an equation for: The value of d dimes is $27.50
WE know that the value of d dimes is 10d cents (from our previous lesson) .. but $27.50 is in terms of dollars so we need to change it to cents . $27.50 is 2750 cents ... So our equation becomes 10d = 2750.

Algebra Honors (Period 6 & 7)

Proof in Algebra 3-8
Some of the properties discussed in the previous chapters are statements we assume to be true. Others are called theorems. A theorem is a statement that is shown to be true using a logically developed argument. Logical reasoning that uses given facts, definitions, properties, and other already proven theorems to show that a particular theorem is true is called a proof. Proofs are used extensively in Algebra as well as Geometry.
Prove: For all numbers a and b, (a + b) – b = a

Many times, only the KEY reasons are states—the substitute principle and the properties of equality are usually not stated. So, the above prove could be shortened to 4 steps:

Prove: For all real numbers a and b, such that a≠ 0 and b ≠0
1/ab = 1/a ⋅1/b

Since 1/ab is the unique reciprocal of ab, you can prove that

1/ab = 1/a⋅ 1/b by showing that the product of ab and 1/a ⋅ 1/b is 1

Once a theorem has been proved, you can use it as a reason in other proofs. Check the Chapter summary on page 88 of our textbook for the listing of properties and theorems that you can use as reasons in your proofs for our homework.