Radical expression

Radical expression

Any mathematical expression containing a radical symbol(√) is called a radical expression. Generally we use the symbol '√' 

to determine the square root of a number. But, this symbol may be used to represent the 'cube' , 'fourth', or higher roots of a number.

Definition:

In mathematics, any expression containing the radical symbol (√) is known as radical expression. 

Let's explain the matter...

  Let, 'n' be any positive integer (n>1) and 'a' be any real number; then the expression, ⁿ√a or  (a)ˡ/ⁿ  is called a radical expression. Here, 'a' is known as radicand and the symbol '√' is known as radical. Here, the form  (a)ˡ/ⁿ  has a special name: "exponent form". 

  Examples:

   √16=4=(16)ˡ/² , √5,√7,...etc.

History:

The term "radical" is derived from a Latin word 'radix'. In Latin 'radix' means 'root'. 

In 1600s radical expressions were first used in England. Then the uses of radical expression spread worldwide.

Properties of radicals:

If a(>0) and b(>0) , then

(1) √a×√b = √(a×b)

(2)√(a/b)=(√a)/(√b)

(3) √(a+b) is not equals to (√a+√b).

(4) √(a-b) is not equals to (√a-√b).

Note:

  The expression (√a+√b) is called the conjugate of the expression (√a-√b). Therefore, the expressions (√a+√b) and (√a-√b) are conjugate to each other. so, we can use the conjugate to rationalize the denominator of a radical expression.

Simplified radical expression:

A radical expression is said to be in simplified radical form,  if each of the following are true:

(1) All exponents in the radicand must be less than the index.

(2) Any exponents in the radicand can have no factors in common with the index.

(3)No fractions appear under a radical.

(4) No radicals appear in the denominator of a fraction.

   As for example, simplyfy: ⁹√(a⁶).

  We have , ⁹√(a⁶) = (a⁶)ˡ/⁹ = (a)⁶/⁹= (a)²/³= ³√a².

  So, simplified radical expression of ⁹√a⁶ is ³√a².

Method to simplyfy a radical expression:

There are many radical expressions , where the radicand is not a perfect square or cubes or higher powers of a number. In such cases to simplyfy the expression , we may use the following method.

  At first factories the radicand in all possible prime factors. Then collect the same prime factors and move them outside the radical sign. Then multiply the factors inside and outside the radical sign separately. The result is in the simplified form.

As for example, simplyfy: √480.

  we have, √ 480=√(2×2×2×2×3×2×5)

                             =2×2√(3×2×5)=4√30.

Some formulas to solve a radical expression:

(1) (aᵐ)×(aⁿ)=aᵐ⁺ⁿ 

(2) (aᵐ)÷(aⁿ)=aᵐ⁻ⁿ

(3) (aᵐ)ⁿ= aᵐⁿ

(4) (ab)ᵐ= (aᵐ)×(bᵐ)

(5)(a÷b)ᵐ= (aᵐ)÷(bᵐ)

(6) If m is a positive number, a⁻ᵐ=1÷(aᵐ). Here, a⁻ᵐ is called reciprocal of aᵐ.

(7) If m,n are integer, aᵐ/ⁿ means (aᵐ)ˡ/ⁿ ; i.e., n-th root of aᵐ.

(8) If m=0, a⁰ is meaningless, a⁰=1.

(9) If a,m,n real and aᵐ=aⁿ then, m=n, where, a not equal to 0,1,-1.

(10) If a, b, m are real , and aᵐ= bᵐ then, either a=b or m=0.

If you find out any incorrect information or know anything more about this , please write it in the comment section!

Permutation and combination

Permutation

Let us arrange three types of fruits namely  Apple, banana and mango in all possible ways(each is different).
   Then we will get  six different varieties.
 Apple, banana, mango;
 Apple, mango, banana;
 Banana, mango, apple;
 Banana, apple, mango;
 Mango, apple, banana;
 Mango, banana, apple;

All are different!

The arrangement of things like this are known as Permutation.

 Definition: 

Permutation is arrangement of things in all possible ways.

 In permutation the order of things is  considered.

     As for example , let, we have to form a number  consisting of three digits using the digits 1,2,3,4 . To form this number the digits have to be arranged  in some order. Different numbers will get formed depending upon the order in which we arrange the digits. In this way , each arrangement of the digits is a permutation.

  Again , let , there are three prizes and nine participants in a competition.

 We are to distribute the prizes among the top three(first, second and third) participants. Then we are to choose three people out of nine. Now, the first winner can be chosen in 9 different ways. The second winner can be chosen in 8 different ways. And  the third winner  can be chosen in 7 different ways. Thus we have total 9×8×7 different ways to choose three winners from a set of 9 participants.

   We know that, 9!=9×8×7×...×2×1.

Now , 9×8×7=9!/6!. That is, 9!/(9-3)!.

   In general, there are n!/(n-k)!  different ways to arrange k elements out of n elements in some order. Generally, it is denoted by P(n,k).

  So, P(n,k) = n!/(n-k)!.

    

Combination

 Let we are to select 11 players out of 15 players to form a cricket team. We can select any 11 of the 15 players randomly. Here if we change the order of the players the team does not changes. So, in combination order is not considered.

Definition:

Combination is the selection of things. 

In combination  the order of things is not considered.

As for example, let we are to distribute 3  prizes(same) to 3 winners(first, second and third) out of 9 participants. Since prize is same for all, the order(first, second, third) does not matter. Now, we can select 3 winners from 9 participants in P(9,3) different ways. But here order is considered. So, if we does not consider the order we have total P(9,3)/3! ways.

  Thus , we can select 3 winners from 9 participants in P(9,3)/3! ways. The order is not considered here. This is a good example of combination. 

  Generally, combination of k elements out of n elements is denoted by C(n,k).

 C(n,k)= n!/{(n-k)!×k!} = P(n,k)/k!.

 This is the basic concept of permutation and combination.

 Some interesting problems on permutation and combination:

Problem 1:

 How many words can be formed using any four letters from the word "SPRITE" ?

    

   Here, the word "SPRITE" contains six different letters. We are to form words choosing four letters out of these six letters in all possible ways. So, we are to choose four letters from six letters in all possible different ways. Therefore, the number of ways is equals to P(6,4) = 6!/(6-4)! =6!/2!=360.

Finally, 360 words can be formed by choosing four letters from six letters of the word "SPRITE".

 Problem 2:

  Suppose, there are 30 students in a class. We are to form  quiz  teams of 5 students from these  30 students. How many different teams can be formed?

Here , we are to form teams of 5 students from 30 students in all possible different ways. So, the possible number of teams are equals to C(30,5) = 30!/{5!×(30-5)!} = 142506.

  

Note:

(1) P(n,r)= P(n-1 , r) +{r× P(n-1, r-1)}.

(2) C(n,r)+C(n,r-1)=C(n+1,r).

(3) C(n,r) ÷ C(n,r-1) =(n-r+1)/r.

(4) C(n,r) =C(n, n-r).

(5) If C(n,p) = C(n,q) then, p+q=n (p not equals to q).

(6) C(n,1)+C(n,2)+C(n,3)+...+C(n,n)=2ⁿ -1.

   

If you find out any incorrect information or know anything more about this , please write it in the comment section!

Even and odd numbers

Even and odd numbers

Let's enter in the world of numbers. There are several categories of numbers like integers(positive integers and negative integers), rational numbers, irrational numbers, whole numbers, fractions and much more. The integers (both positive and negative) can also be categorised as Even numbers and odd numbers. 

   So, Let's start with even and odd.

  Definition of Even and odd numbers:

An integer number (positive or negative) which is   divisible by two(2) is called an even number.

On, the other hand, an integer number (positive or negative) which is not divisible by 2 is called a odd numbers.

 So, it is very clear that, an even number is not a odd number and a odd number is not an even number.

  As for example,  8 is  divisible by 2 .So it  is an even number. But,  9  is not divisible by 2 . So, it is odd  a number.

The even and odd numbers can be defined in an another interesting way.

  An integer is called an even number if it has no remainder(or, remainder is 0) when it is divided by 2 . Similarly, an integer is called a odd number if it has remainder '1' when  it is divided by 2.

 As for example, 6 has no remainder when it is divided by 2. So, it is an even number.

But, 5 has remainder '1' when it is divided by 2. So, 5 is a odd number.

In  a  more formal way we can define even and odd numbers.

 A number  m  is called an even number if it can be expressed as , m= 2n such that n is an integer.

 Again, a number  k is called a odd number if it can be expressed as , k=2n+1 such that n is an integer.

 Here, 4 can be expressed as, 2×2 +0. So, it is an even number. And, 7 can be expressed as, 2×3 +1 . So, it is a odd number.

Basic operations with even and odd numbers:

(1) even number+even number=even number.

Example: 12+6=18.

(2) even number+ odd number =odd number.

Example:  24+5=29.

(3) odd number+odd number=even number.

Example:  5+7=12.

(4) even number - even number= even number.

Example: 8-4 =4.

(5) even number - odd number= odd number.

Example: 10-5=5.

(6) odd number - odd number = even number.

Example, 7-5=2.

(7) even number × even number= even number.

Example: 8×4=32.

(8) even number × odd number = even number.

Example: 4×7=28.

(9) odd number× odd number= odd number.

Example: 5×3=15.

Zero is an even number.

Yes , it's true. Zero is an even number.

Here is the proof.

 Zero is divisible by 2. Also, 0 can be expressed as, 0= 2×0.

So, it is clear that, 0 is an even number.

Here are some even and odd numbers.

 Even numbers: 0,2,4,6,8,10,12,14,16,18,20,...

 Odd numbers: 1,3,5,7,9,11,13,15,17,19,...

   

 If you find out any incorrect information or know anything more about this , please write it in the comment section!