This week in PreCalc, we expanded on the ideas of radicals by learning how to add, subtract, multiply, and divide them. A lot of the ideas were straight forward, but the most challenging is dividing radicals.

Example:

Work:

Step 1: simply if necessary

Simplifying the given equation if it can be simplified makes the work easier later on because the numbers will be smaller.

Step 2: identify the conjugate

For dividing by a binomial, you have to multiply the numerator and the denominator by the denominators conjugate (is formed by changing the sign between two terms in a binomial). So the binomial in my example has a negative sign, but it’s conjugate will have a positive sign, and that’s the only difference between conjugates. The rest of the terms stay the same.

Step 3: multiply top and bottom by conjugate

Start with expanding the top and the bottom.

Step 5: simplify where necessary

After expanding, combine like terms, simplify radicals, and solve until you no longer can. When you can’t go any further, then you have reached your answer.

This week in PreCalc 11, we learned about Absolute Values (of a real number, is the square root of the square root of a number. In simpler form, it’s the distance of a number on the number line from 0, and can only be a positive. So, the absolute value of -5 would be 5) and Roots/Radicals (a radical expression is defined as any expression containing a radical (√) symbol. It can be used to describe a cube root, a fourth root, or higher depending on the what the index is. The index is the number on the top left corner outside the radical symbol. A root is, when multiplied by itself is given a number. So, 2 is the root of 4 because 2×2=4)

The things we did were pretty straight forward this week, but the hardest thing I came across was evaluating expressions that had absolute values in them.

Example:

Explanation:

Step 1: find the absolute value first

I personally do this first because when the number you’re trying to figure out the absolute value for is a negative, then it turns into a positive so it’s easier to get that out of the way first. In this example, the value of a positive so we just keep it a positive

Step 2: finish the rest of the expression

The rest is simple BEDMAS, and trying to find the answer. Don’t forget, the math in the brackets should be done first, and then the rest follows.

This week in Precalc 11 we learned about Geometric Sequences (patterns that multiply by the same number to get the next number), Finite Geometric Series (patterns that don’t have numbers that eventually start to look the same), and Infinite Geometric Sequences (patterns that never have an ending, but still possible to find the sum because eventually the numbers in the sequence become very close together). The one thing that stood out to me is finding the finite geometric series.

When given a geometric sequence, just like an arithmetic sequence, it’s possible to find the sum of it. Finding it for an arithmetic sequence, though, is different than finding it for geometric sequence. For example, if you need to find of a geometric series you don’t need to find first like for an arithmetic series. Instead, you can just plug in what you know into the equation given.

Ex.

Explanation:

Step 1: Find the common ratio

Common ratio is different than the common difference that we were introduced to in the arithmetic unit. Instead of finding the number that the numbers go up by each time, we find the number that is used to multiply the last number to get the next number.

Step 2: Find

Once we know what the common ratio is, then we have everything we need to fill in the equation to get the answer we are looking for.

This week in Math 11 we learned about Series and Sequences. Sequence is essentially to just a pattern that starts at a number and continually goes up my the same number each time. And a Series is the sum of all the numbers of the sequence . The one thing that stood out for me and was challenging to learn was figuring out how to find being given and . After figuring it out, it’s safe to say it’s essentially just figuring out what you don’t know, what you do know, and which equation to use.

Example:

The Work:

Step 1: Figure out what you know and don’t know based on what’s given in the question

In this example, we know and and we don’t know what or is equal to or what is equal to.

** is the first number in the sequence the series represents

** or or any number that is attached to the is the general term that represents the place of whatever the number is ( is the 15th term in the sequence)

**  is the difference between the number add uns in the pattern which has to stay the same throughout to be a artithmetic sequence

Step 2: Find d

Finding D is the most important in this question because before we can find we have to find and we can’t do that without knowing the difference between each term in the pattern

Step 3: find

Finding the 30th term is important too because of its connection to which is that for there to be an there has to be a 30th term in the sequence that the series can add up until. And to figure out we need to know what is because we need to know what the number is to add in the series

Step 4: Figure out what you know now and what you still need to know

In this example, now we know at this point that , , and  and we still don’t know what is equal to

Step 5: Find

At this point, we have all the information we need to use the series equation and find what we were looking for in the beginning