### February 2018 archive

When dividing radicals, the denominator can never be a radical. So in order to solve, the denominator must first be made into a rational number.

First, use a difference of squares to rationalize the denominator. But since whatever you do to the bottom you must also do to the top, multiply both the numerator and denominator.

now add like terms to get a more simplified expression.

now you can just simplify the expression as far as you possibly can! It’s okay for the expression to not equal a single number.

The absolute value of a real number is defined as the principle square root of the square of a number.

for instance the absolute value of $\mid 4 \mid$ is 4

when a number is between the brackets, it must be brought out of the bracket like this:

$\mid 4 \mid$

$=4^2$

$=\sqrt {16}$

$=4$

Whenever a number is between brackets, the answer must be a positive number. However, after the number is out of the brackets it can become negative in an equation.

We also learned about roots and radicals. Numbers with square roots can never be negative, like $\sqrt {-25}$ because it cannot be calculated and results in an error when put into a calculator. On the other hand, a cubed number can be negative. Roots with an index that’s even cannot be negative, but roots that have an uneven index can be negative.

when simplifying roots that are not perfect, a different approach must be taken.

for example:

$\sqrt {24}$

$=\sqrt {4*6}$ find two multiples of a number, one being a perfect square

$=\sqrt {4} * \sqrt {6}$ separate the roots

$=2 \sqrt {6}$ square the perfect square and leave the radical as a square root

this applies to other index’s as well, just find perfect cubes instead of perfect squares, etc.

This week we learned about geometric sequences and series. A Geometric sequence differs from an arithmetic sequence because instead of adding a common difference, a common ratio is multiplied by the term to get the next term.

the equation used to find a term in a sequence is

$t_n=ar^{n-1}$

in this equation $t_1$ is represented by $a$

for example: 5,10,20,40… find $t_{10}$

$t_{10}=5(2)^{10-1}$

$t_{10}=5(2)^9$

$t_{10}=5(512)$

$t_{10}=2560$

A finite geometric series can be found using the equation

4, 9, 14, 19, 24

$d=5$, $t_1=4$

$t_n=t_1+(n-1)d$

$t_n=4+(n-1)5$

$t_n=4+5n-5$

$t_n=5n-1$

using this information, we can find any value of $t$ using the equation $t_n=5n-1$. We can plug in any term into $n$ to find $t_n$

this equation will work for any term within the sequence, as long as the starting term and common difference are constant. If either of these change it is no longer an arithmetic sequence.

To find $S_{50}$, the sum of all 50 terms, we must use the equation $S_n={n}{2}(a+1)$

In this equation, $n=$ number of terms(50) and $a=$ the first term(4)

$S_{50}={50}{2}(4+1)$

$S_{50}=25(5)$

$S_{50}=125$