# Week 4: Rationalizing the Denominator

Hey, this week in Pre Calc 11 I learned something new that is difficult for me to understand. by me teaching this to you we both have a chance to learn more about rationalizing the denominator.

How to Rationalize the Denominator

ex. $\frac{1}{5\sqrt{3}}$

Step One: Multiply by the denominator

Step Two: Root the Radical Expression in the Denominator

Step Three: Simplify Further if Possible

FINAL ANSWER IS $\frac{\sqrt{3}}{15}$

Now you know how to remove a radical expression from the denominator. 🙂

# Week 3 Absolute Value and Simplifying Radical Expressions

This week in Pre Calculus 11, we learned what an absolute value of a real number was and how to simplify radical expressions.

Absolute Value of a Real Number : The principal square root of a square number.

SOLUTION: The absolute value of a negative number is the opposite number, and the absolute value of a positive number and 0 are the same.

ex/    |-99| = 99         | 12 | = 12

This is because distance is always positive.

The long lines act as brackets but are not and it represents absolute value.

• ex 1/         4 |15-20|
•                     4 |-5|
•                     4 (5)
•                     = 20

• ex 2/         |6 +(-10)| -|5-7|
•                     |6-10| – | -2|
•                     |-4| – |2|
•                     4  –   2
•                     = 2

Here is a video that really helped me learn and understand a little bit more about absolute value.

we also briefly reviewed radical expressions

ex/ $\sqrt{25}$ =   5  and $3\sqrt[2]{5}$ = $\sqrt{45}$

Solving

Step 1: Find perfect squares.

Step 2: Take them out / simplify

Step 3: Do the same with the variables (treat them like numbers)

ex/ $\sqrt[2]{18x}$

• ex/ $\sqrt[2]{18x^2}$
• $\sqrt{9\cdot{x^2}\cdot2}$
• $\sqrt{3\cdot3\cdot{x}\cdot{x}\cdot2}$
• $3x\sqrt[2]{2}$

This is how to work with variables in Radical Expressions

# Week 2 Geometric Series

Week 2: This week in Pre Calculus 11 we learned about infinite and finite geometric series.

GEOMETRIC SEQUENCES – Each term is multiplied by a constant, known as the common ratio.

There are two types of infinite geometric series. (adding the sequence is a series to find a sum).

DIVERGING – A diverging series means the ratio is    r > 1   or    r < -1. This means the number of terms will increase, which means the partial sum increases, so the series does not have a finite sum. (NO SUM) It only has a sum if it is asking for a specific sum of terms.

CONVERGING – A converging series means the ratio is 1 > r > 0 (smaller than 1, bigger than 0) or -1 < r < 0. ( bigger that -1, smaller than 0). The partial sum will appear to get closer to a number, therefore we estimate the finite sum. (HAS SUM).

To find the sum:

1.  Find Ratio
2. Identify if it Diverges or Converges.
3. Diverges, STOP (No Sum)
4. Converges, CONTINUE (Sum)
5. Use the formula $S_{\infty} = \frac {a}{1- r}$

ex. 8 + 16 + 32 + 64 + 128 + 256 …

Ratio: $\frac {16}{8} = 2$

Identify: Diverges. NO SUM

ex. 2)     8 + -7.2 + 6.48 + -5.832 + 5.2488 …

Ratio: $\frac {-7.2}{8} = -0.9$

Identify: Converges, ratio is a decimal bigger that -1 but smaller than 0.

FORMULA :

• $S_{\infty} = \frac {a}{1 - r}$
• $S_{\infty} = \frac {8}{1 - (-0.9)}$
• $S_{\infty} = \frac {8}{1 + 0.9}$
• $S_{\infty} = \frac {8}{1.9}$
• $S_{\infty}$ = 4.21052…
• Estimated sum = 4.2

This is how to find the sum of a geometric series.

If a question asks for the sum of a term the formula to use is:

• $S_{n} = a \frac {(r^n -1)}{r - 1}$

ex. 8 + 16 + 32 + 64 + 128 + 256 …

Find $S_{10}$ :

• $S_{10} = a \frac {(r^n -1)}{r - 1}$
• $S_{10} = 8 \frac {(2^{10} - 1)}{2 - 1}$
• $S_{10} = 8 \frac {(1024-1)}{2-1}$
• $S_{10} = 8 \frac {1023}{1}$
• $S_{10} = 8 (1023)$
• $S_{10} = 8184$

# Is Protesting Efficient?

The idea of politics can be very perplexing. The government may be in charge, but that doesn’t give them the right to neglect the needs and wants of the citizens in that country. The government is required to listen the people, unfortunately, the voice of the people is not always heard. Situations may be dealt with by civil disobedience, creating petitions, and even hiring lobbyists. These can have positive or negative impacts. On the most part, these are very efficient ways to overcome conflicts and cause movement for change.

Civil disobedience is typically an efficient way to attract attention to a problem. Civil disobedience is generally a person who accepts the consequences of breaking a law that they most likely, find absurd. For example, Viola Desmond, refused to sit in the designated colored area in a theatre in 1946. The situation escalated quickly, and she was then jailed for not listening to the rules of a theatre. This received attention and was advertised in local news paper and decades later a book was published, causing change with racial discrimination. The benefits of civil disobedience is, the situation draws attention, there is an increase in chance of change, and it is a non-violent way of protesting. Despite civil disobedience being efficient, there are cons to it. This way of protesting may result in jail time, causes a tougher resistance towards the issue, and may take some time to make change. In the end change may not occur, but an issue is being approached and it causes awareness. Civil disobedience works on the most part, depending on the situation that needs attention.

Creating petitions is another great way to voice opinions. Petitions are an easy way to show that many people have common feelings towards an issue. Change is typically made because it is a very formal and non-violent way to get a point across. For example, in 2008 HST was introduced and it replaced PST and GST. Many people found it was detrimental and was not placed to benefit the people. It took a few years to revert to PST and GST, but it eventually happened. The cons to petitions are, it may take some time, the issue must be serious, and it can be challenging to find the minimum number of people that share the same feelings to get the problem noticed. For the most part if the issue is serious the government should step in to keep the people happy and to keep a good reputation. Otherwise, the issue will be dismissed and left unresolved.

Lobbyist and pressure groups are very productive. Lobbyist are generally hired to persuade the government to see and make changes according to the people. NAFTA is an example that used pressure groups to enforce free trade. Benefits of using pressure groups is they use a democratic process, they daunt the majority as well as protect individuals and give them a voice to suggest and give their opinions. They are also able to persuade officials. Moreover, they can provide positive solutions, because of the skill and knowledge of being a lobbyist. Lobbyists are usually hired because they have many connections in which they can talk to, to get their points across. Consequently, pressure groups could lead to a higher resistance towards the issues, the breaking of laws may occur, and it is sometimes only beneficial to one side of the issue. Again, pressure groups are very productive but are not always the best choice of making change.

In the end, there are many ways citizens can go about to make change. Civil Disobedience, petitions, and lobbyists are effective ways to voice the need for change. These methods have been used for decades and still work today, but to each benefit there will be a negative effect from protesting. Governments may or may not fix an issue, but they are sure to approach one if it brings enough attention. Governments relatively like to keep people happy and keep a good reputation. If an issue occurs, be sure to know the pros and cons to each protesting method to ensure change.

# Week 1 Arithmetic Sequences

This week was the first week of Pre Calculus 11 for me. So far it has been good, as we are learning about series and sequences. Today I am going to teach you the difference between arithmetic sequences and an arithmetic series. I will also show the formula to find the nth-term and how to find the sum of the terms. STAY TUNED 🙂

A SEQUENCE for example, is a set of numbers that are changing in some way.

ex/ 5, 10, 20, 40, 80

5 is called $t_1$ , 10 = $t_2$ , 15 = $t_3$ and so on.    The t stands for term.

An ARITHMETIC SEQUENCE is a sequence that changes by a constant amount. The constant amount is also known as the common difference.

ex/ 5, 10, 15, 20, 25,     d=+5 ;The common difference is +5

to find the common difference the formula is  $t_2$  –  $t_1$

NOT ARITHMETIC,       NO CONSTANT DIFFERENCE

An ARITHMETIC SERIES is the terms of an arithmetic sequence added together. The point is to find the sum of the desired terms.

ex/ 5+10+15+20+25 = 75

How to find the nth-term

Say you wanted to find what $t_{50}$ is in the sequence 12, 9, 6, 3, 0 is but you don’t want to continue writing the whole sequence out. Well your in luck, there is a fast way.

The formula to find the nth-term is  $t_n$ = $t_1$ + d(n-1)

• The nth-term is $t_n$$t_{50}$
• d= -3
• $t_1$ = 12

Now we insert all the known numbers into the formula

1. $t_n$ = $t_1$ + d(n-1)
2. $t_{50}$ = 12 + (-3)(50-1)
3. $t_{50}$ = 12 + (-3)(49)
4. $t_{50}$ = 12 -147
5. $t_{50}$ = -135

There you are, you just figure out how to find the 50th term quick and easy.

How to find the sum of a series

We will take the example from above and determine the sum of  $S_{50}$$S_{50}$ means $t_1$ to $t_{50}$ will be added.

The formula to determine the sum is $S_n = \frac {n}{2}(t_1 + t_n)$

• $S_n$$S_{50}$
• n = 50
• $t_1$ = 12
• $t_n$ = -135

Insert into formula

1. $S_n = \frac {n}{2}(t_1 + t_n)$
2. $S_{50} = \frac {50}{2}( 12 + (-135))$
3. $S_{50} = 25( -123)$
4. $S_{50} = - 3,075$

There you have it. Today you learned what an arithmetic sequence and series were, what the formula to find the nth-term and the sum of a series was, and how to solve. I hope this blog post helps you in your future with practice and studying.