Concept of Simplification of Radicals

Various concepts are used in mathematics. Radicals are also one among them. They can also be represented with the help of a symbol. Basically they are used to represent the root of a number. The number can from the various number systems in mathematics. There are various number systems in mathematics.

The natural number system, whole number system, integer number system and real number system are some of them. There can also be irrational numbers and also complex numbers. The complex numbers have a real part and an imaginary part attached to them.

Do they are said to be complex numbers. The operations that are performed on real numbers like addition and subtraction can also be performed on the complex numbers as well. But to perform these operations one must be clear with the concept of the complex numbers.

The next question that arises is how do you simplify radicals in mathematics. The radicals have to be simplified in order to arrive at the final answer. The process to simplify radicals is quite simple. The root that has been applied to a number can be a square root, cube root and so on.

Sometimes fourth root can also be used. The number inside the square root symbol must be present two times then it can be taken out of the square root symbol. In case of cube root symbol it should be three times and in the case of the fourth root it should four times and so on.

In simplifying radicals practice is required; otherwise it becomes very tough to solve the problems. In mathematics practice is very important otherwise it will be very difficult to arrive at the final answer. Many mistakes can also be committed en-route to the answer. This can be avoided by proper practice.

There can questions to simplify radicals with variables like the simplification of normal radicals. The process is quite similar to the process of simplification of the radicals of the normal nature. Only proper practice can help one to reach at the right answer. There can be signs that can be used before the radical symbol.

The signs can be positive or negative in nature. Depending on the sign preceding the radical symbol the answer also changes. The positive denotes a positive number and a negative sign denotes a negative number. The magnitude also plays a very important role.

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BODMAS

BODMAS is one of the most important concepts in mathematics. Taught in the middle school, BODMAS plays a major role throughout every mathematical calculation. BODMAS is an abbreviation of rules of operations in mathematics. It is used to make the rules easy to remember and perform. BODMAS is referred by various other names. BODMAS is also known as BEDMAS where E means Exponents. It is also known as PEMDAS where P means Parenthesis. Let’s have an in depth look at the full-forms of BODMAS and its meaning in this post.

B – Brackets First
O – Orders such as powers, square roots etc.
DM – Division and Multiplication
AS – Addition and Subtraction
• BODMAS defines the rules to be followed in orders of operations. According to the rule, the operation within brackets should always be calculated first.
Example: (2 toys for infants x 3 kids) + (2 crib toys for baby girls x 2 girls) + 5 toy action figure for kids
6 toys for infants + 4 crib toys for baby girls + 5 toy action figure for kids
15 toys for kids
• According to the BODMAS rule, exponents such as power and roots should be calculated prior to multiplication, division, addition and subtraction.
Example: 5 infant toys x 22
= 5 infant toys x 4
= 20
• According to the BODMAS rule, multiplication and division should be performed prior to addition and subtraction.
Example 1: 2 + 5 x 3
= 2 + 15
= 17
Example 2: 45 – 16 / 4
= 45 – 4
= 41
• According to the BODMAS rule, addition and subtraction comes at last when there are other operations to be performed.
Example 1: 2 + 3 x 6
= 2 + 18
= 20
Example 2: 7 – 2 x 3
= 7 – 6
= 1
These are the basics about BODMAS.

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Factoring polynomials

When factoring out polynomials,  we find   the  polynomial  that divide out evenly from the original polynomial  .   How to factor  Polynomials:  For this we have to find all the terms that if multiplied together we get the  original polynomial.  This is continued to all the terms until this cannot be simplified any more.  If the polynomial cannot be factored any more then the polynomial is said to be completely factored.


A factor of a polynomial is any polynomial which divides evenly into   the given polynomial   For example, x + 2 is a factor of the polynomial x^2 – 4.
The factorization of a polynomial is its representation as a product its factors. For example, the factorization of x^2 – 4 is (x – 2)(x + 2).

In mathematics, factorization or factoring is the breaking apart of a polynomial into a product of other smaller polynomials. If you choose, you could then multiply these factors together, and you should get the original polynomial.
This example shows how to factoring polynomials.    Take 3x^2 – 12x + 9.   The common term in these is 3.  So take 3 out and divide each term by 3.  We get   3 (x 2-4x + 3).

Step by step explanation as how to Factor out Polynomials is given below
Factor 4a^2 + 20a – 3a - 15


The first two terms have a common factor in 4a.  The last two terms have a   common factor in 3.
We need to factor those terms out.

4a ( a + 5)  -3(a+5)

Now you have a binomial.  Each term     has a factor of (a + 5).

(4a -3) (a+5) is the factored terms.


Another example is given below on factoring of polynomials
X2 -8x + 15

start by looking at the factor pairs of 15.  We are looking for a pair of factors which add up to equal -8.  Look for the factor pairs of  +15 so that they add up to -8    The negative factor pairs of 15 are:
-15  and  -1
 -5 and  -3
Since -5 + -3 = -8 this is the pair we are looking for and we can factor the
original expression into:  (x-5)(x-3)

Matrix types: Diagonal matrix

The term diagonal matrx refers to the topic of linear algebra which is a branch of mathematics. In general a matrx would look like as follows:
[a11 a12 a13 … … a1m]
[a21 a22 a23 … … a2m]
[a31 a32 a33 … … a3m]
[… … … … … …    ]
[… … … … … …    ]
[an1 an2 an3 … … anm]

The above matrx has n rows and m columns. The diagonal of such a matrx consists of all the entries where in the row number = column number. Therefore if r = row number and j = column number. Then the entries of the type a(ij) where i=j are the diagonal entries. Therefore in the above matrx the diagonal would be the highlighted entries as shown below:

[a11 a12 a13 … … a1m]
[a21 a22 a23 … … a2m]
[a31 a32 a33 … … a3m]
[… … … … … …    ]
[… … … … … …    ]
[an1 an2 an3 … … anm]

Now if we have a matrx where in all the entries except the diagonal entries are zero, then such a matrx would be called a diagonal matrix. In general a diagonal-matrx would look like this:

[a11 0 0 … … 0]
[0 a22 0 … … 0]
[0 0 a33 … … 0]
[0 0 0 … … 0]
[… … … … .. 0]
[0 0 0 … … ann]

Note that a diagonal-matrx has to essentially be a square matrx. An example of a 3x3 diagonal-matrx is shown below:

[2 0 0]
[0 -1 0]
[0 0 5]

Scalar matrix:

A diagonal-matrx in which all the entries of the diagonal are equal is called a scalar-matrx. The general form of a scalar matrx would be like this:
[a 0 0 … … 0]
[0 a 0 … … 0]
[0 0 a … … 0]
[0 0 0 … … 0]
[… … … … .. 0]
[0 0 0 … … a]

Determinant of diagonal matrix:
Let us try to understand how to calculate the determinant of a diagonal matrix using an example.

Example: Calculate the determinant of the following diagonal-matrx:
[2 0 0]
[0 -1 0]
[0 0 5]
Solution:
D = 2*(-1*5 – 0*0) – 0*(5*0-0*0) + 0*(0*0-(-1)*0))
= 2*(-1)*(5) – 0 + 0
= 2*(-1)*5 = -10
So we see that the value of determinant of a diagonal-matrx is the product of the terms on the diagonal (also called the principal diagonal)