Week 4 : Simplifying And Adding Radicals

Hi guys,Today i will tell you about the Simplifying and Adding radicals.

So let’s start with the first example :

Adding : 2\sqrt{11} + 8\sqrt{11}

So we can know : the two radicals inside the roots. they have same the number “11”. This means you can combine them but only combine the number before the roots. We will still keep the radicals inside roots

Let’s go to another example :

8\sqrt{5} + \sqrt{3} + 3\sqrt{5} + 10\sqrt{3}

So with this example we can not adding with diffirent radicals. First, I always sort the number have same radicals are next to each other. Then we will adding each number have the same radicals. We will get like this :

8\sqrt{5} + 3\sqrt{5} + \sqrt{3} + 10\sqrt{3}= 11\sqrt{5} + 11\sqrt{3}

Just remember, never do combine the number have diffirent radicals like
8\sqrt{5} + \sqrt{3} and then get 9\sqrt{8} . It is the big mistake if you combine it together.

This is not correct because \sqrt{5} and \sqrt{3} are not same radicals so we can not be added them together.

Week 3 : Absolute Value Of A Real Number – Simplyfying Radicals

The absolute value of a number is its distance from 0
Example : The absolute value of 3 or -3 is 3 because they have same distance from 0 and absolute value of 3.

The symbol for absolute value is a bar |∣vertical bar on each side of the number.

Example : The absolute value of 25, we can write it like this : |25|=25

Simplyfying Radicals :

A radical is a number that has a fraction as its exponent.

Example :

\sqrt{20}= \sqrt{4.5}
\sqrt{20}=2\sqrt{5}

\sqrt{20} : Intire
2\sqrt{5} : a mix of whole numbers and radicals

Another example : \sqrt{75}= \sqrt{25.3}
\sqrt{75}=5\sqrt(3)

Ex: \sqrt[3]{96}= \sqrt[3]{16.6}=2\sqrt[3]{12}

\sqrt[3]{-81}= \sqrt[3]{-27.3}=-3\sqrt[3]{3}

Week 2 : Geometric Sequence

Geometric Sequences
In a Geometric Sequence each term is found by multiplying the previous term by a constant.

Example : 2,6,18,54, 163,486,1458,…

This sequence has a factor of 2 between each number

Each term is found by multiplying the previous term by 3 ( except the first term : 2 )

Infinite Gemetric Series :

In Infinite Geoemtric Series, when : r>1 or r<-1 it will be converges Let's see my example, and see what happends : 1,1/3,1/9,1/27,.. We have: a = 1 (the first term) r = 1/3 (halves each time) And so use this formula : [latex]S_ \infty[/latex] [latex]S_ \infty[/latex]=1/1-1/3 [latex]S_ \infty[/latex]=2/3 (0,666666..)

Week1 : Arithmetic Sequences

t_n t_{50} S_n=\frac{n}{2}(t_1+t_n)

In an Arithmetic Sequence the difference between one term and the next is a constant.

In General we could write an arithmetic sequence like this:

{at, t+d, t+2d, t+3d, … }

where:

t is the first term, and
d is the difference between the terms (called the “common difference”)
Example : My Arithmetic Sequences : 11,2,-7,-16,-25,-34

Know :
– t = 11 (the first term)
– d = -9 (the “common difference” between terms)

We can write an Arithmetic Sequence as a formula :
t_n= t1+(n-1).d

Using the Arithmetic Sequence formula:

tn=t1+(n-1).d
tn=11+(n-1).(-9)
-34=11+9-9n
-34=20-9n
n=(6)

Example 2 : I want to determine t50
I know : t1= 11
n=50
n-1=50-1=49
Using the Arithmetic Sequence Rule :

t_{50}= t1+49.d
t_{50}= 11+49.(-9)
t_{50}= -430

To sum up the terms of this arithmetic sequence:

t1 + (t2+d) + (t3+2d) + (t4+3d) + …

Use this formula: S_n=\frac{n}{2}(t_1+t_n)

Example: Determine the sum of this Arithmetic Sequence : 11,2,-7,-16,-25,-34

The values of t,d and n are:

t = 11 (the first term)
d = -9 (the “common difference” between terms)
n (how many terms to add up)

S_n=\frac{n}{2}(t_1+t_n)
S_n=\frac{6}{2}(11+-34)
S_n=(3.-23)
S_n=(-69)