Radical is a Latin word that has the meaning of roots and sometimes it is also known as radix in mathematics. Radicals are used to define the root of the algebraic function or other Functions of abstract Algebra. Radicals Math are used for many different type of concepts as linear algebra or number theory or different types of modules etc. These are denoted using the symbol of √ (radical symbol) or radicals define the principal of nth of the value or expression S. Here s is the radicand that should be a real number and n is defined as the index that is always a positive Integer that is greater than one. When Radicals are simplified then it is shows in the way that in the radicals there is fraction left and In the radicand there is no factor of perfect power and radicand S have no exponents and also that is greater than the index n. It should also be noted that if answer is a fraction then there is no radicals appear in the denominator .There are several examples related to the radicals as follows:
Examples: √ 33 (simplest form) or
√ 9 . √ 2 or 2. 3 √ 6 etc.
There are many properties of radicals that are define to understand the nature of the radicals as :
Property no ( 1 ) : n √ k n = k .
Property no ( 2 ) : n √ p . k = n √ p . n √ k .
Property no ( 3 ) : n √ p / k = n √ p / n √ k .
Examples for explain these properties :
3 √48 = 3√ 6 . 8 = 3√ 6 . 3√ 8 = 2 . 3√ 6 .
Square Roots of Whole NumbersBack to Top
a2 = x, which means that the square of a is equal to x, then we can say
√x = a, which represents that square root of x is equal to y.
As we know that if we multiply 1 by 1, we get 1. So we can write:
12 = 1 * 1 = 1
Or 12 = 1
Thus if we reverse the relation, we get √1 = 1
Let us see the squares of other whole numbers, which will help in finding out the square roots of different numbers.
Square of 2 = 22 = 2 * 2 = 4
Square of 3 = 32 = 3 * 3 = 9
Square of 4 = 42 = 4 * 4 = 16
Square of 5 = 52 = 5 * 5 = 25
Square of 6 = 62 = 6 * 6 = 36
Square of 7 = 72 = 7 * 7 = 49
Square of 8 = 82 = 8 * 8 = 64
Square of 9 = 92 = 9 * 9 = 81
Square of 10 = 102 = 10 * 10 = 100
In the same way , we can proceed for all Natural Numbers.
Now if we reverse the above process, then we will get the square roots of whole numbers, so we can write
√1 = 1
√4 = 2
√9 = 3
√16 = 4
√25 = 5
√36 = 6
√49 = 7
√64 = 8
√81 = 9
√100 = 10
√121 = 11
√ 144 = 12
We can apply this process for other numbers too.
Besides these numbers, we have certain other numbers like 2, 3, 5, 6, 7, 8, 10 etc. which are not the perfect squares. These numbers also have the square root, but it comes in Decimals and can be calculated by the division method. These square roots are mostly Irrational Numbers and they cannot be expressed in the form of Fractions. We must also remember that the square roots of negative numbers cannot be calculated and are expressed as complex numbers.
Cube Roots of Whole NumbersBack to Top
y3 = x, which means that the cube of y is equal to x, then we have
3√x = y, which represents that cube root of x is equal to y.
As we know that if we multiply 1 by itself 3 times, we get 1. So we can write that
13 = 1 * 1 * 1 = 1
Or 13 = 1
Thus if we reverse the relation, we get 3√1 = 1
Similarly we can find other cube roots of whole numbers, which will help us to find the cube roots of different numbers.
Cube of 2 = 23 =2 * 2 * 2 = 8
Cube of 3 = 33 = 3 * 3 * 3 = 27
Cube of 4 = 43 =4 * 4 * 4 = 64
Cube of 5 = 53 = 5 *5 * 5 = 125
Cube of 6 = 63 = 6 * 6 * 6 = 216
Cube of 7 = 73 = 7 * 7 * 7 = 343
Cube of 8 = 83 = 8 * 8 * 8 = 512
Cube of 9 = 93 = 9 * 9 * 9 = 729
Cube of 10 = 103 = 10 * 10 * 10 = 1000
Cube of 11 = 113 = 11 * 11 * 11 = 1331
Cube of 12 = 123 = 12 * 12 * 12 = 1728
And in the same way, we can proceed for all Natural Numbers.
Now if we reverse the above process, then will get the cube roots of any number, so we will write
3√1 = 1
3√8 = 2
3√27 = 3
3√64 = 4
3√125 = 5
3√216 = 6
3√343 = 7
3√512 = 8
3√1000 = 10
3√1331 = 11
3√ 1728 = 12 and this way this process proceeds for other numbers too.
Besides these numbers, we have other numbers which are not perfect cubes. Cube roots of such numbers are Irrational Numbers and they cannot be expressed in the form of perfect ratios. We must also remember that the cube roots of negative numbers cannot be calculated and are expressed as complex numbers.
Operations with Radical ExpressionsBack to Top
Different operations on radical expressions can be performed. Some of them are as follows:
1. ( n√a )n = a, this means if we write ( 3√4 ) 3 = 4 . It represents that Cube of cube root of any number is the number itself.
2. n√a * n√b = n√( a * b ), which means if we have = 3√2 * 3√5
= 3√( 2 * 5 )
It represents that if the cube root of 2 and cube root of 5 are multiplied, then it will be equal to the cube root of the product of 2 and 5 i.e. cube root of 10
3. n√a / n√a = n√( a / b ) , which means that if we have = 3√2 / 3√5
= 3√( 2 / 5 )
It represents that if the cube root of 2 is divided by cube root of 5 , then it will be equal to the cube root of the quotient of 2 and 5 i.e. cube root of 2 / 5.
4. Finding the conjugate radical is another operation with radical expression, which means two Radicals which only differ by the sign of + or – are called conjugate radicals. Example of conjugate radicals: √2 + √3 and √2 - √3.
5. While performing operations with radical expressions, we must always remember that only radicals with similar order can be multiplied or divided and only similar radicals can be added or subtracted. If we have the radicals which are not of same order and we need to perform the Operation of Multiplication or division on them, then we should first reduce them to the same order and then perform the multiplication or division operation.
If the product of the two radicals is taken out and we get a rational number as the product, then each of the radical is called the rationalizing factor of the other radical.
Suppose we have √63 * √7 = √441, which is the Square root of 21 and we know that 21 is the rational number and not the irrational number, although the two numbers √7 and √63 are irrational.
Square Roots of Monomial Algebraic ExpressionsBack to Top
36 = 2 * 2 * 3 * 3,
Now to find the square root of 36, we pair the factors and write it one time. So we can write
Root (36) = root (2 * 2 * 3 * 3),
Or root (36) = 2 * 3 = 6.
Now we look at the variables. Word square root means the power of ½. So we need to find the ½ power of the variable for which we need to find the square root.
If we have x6, and we need to find the square root of x6, then we write it as (x6)½.
So it becomes = x3,
It can also be calculated as follows
We can write: x6 = x * x * x* x *x * x,
Now Root (x6) = root (x * x * x* x *x * x),
Now we can replace 2 x’s with one, in order to find the square root of x6. So here we observe that 6 x’s can make 3 pairs.
So x6 = x3,
In the same way if we have to find Root (y8) = (y8)1/2 = y4,
In case we have the monomial expression with the odd number of power, in this case we will get the resultant square root, which is in form of the square root. So if we write z3, then its square root will be
root (z3) = root (z * z * z),
= z * root (z).
In the same way if we have 4 * y2 and we want to find its square root, and then we write:
Root (4 * y2) = root (2 * 2 * y * y) = 2y.