Radicals are numbers left in square roots, cube roots etc.. Radicals are also known as Surds. The root symbol is also called a radical sign and the number and the term or the expression which is under consideration is called a radicand. A radical is a root sign.

Radicals always needs an index to define what root it denotes. For square roots, the two is not written , simply a root sign means that is a square root. For cube roots index is three and the 3 is written on top of the left side of the radical.

The index tells how many times the root must to be multiplied by itself to give the value equal the radicand.In generalize the n^{th} root will be a number multiplied by itself n
times to be equal to the radicand.

**Rules of solving radicals:**

- The denominator of any expressionof the form `a/(sqrt(b)+sqrt(c))` can be rationalised by multiplying the numerator and denominator by `sqrt(b)-sqrt(c)` , the surd conjugate to the denominator,

Similarly, in case of a fraction of the form `a/(sqrt(b)+sqrt(c)+sqrt(d))` where the denominator involves three quadratic radicals, we can rationalise in two operations.

- Multiply both denominator and numerator by `sqrt(b)+sqrt(c)-sqrt(d) ` and then the denominator becomes `(sqrt(b)+sqrt(c))^2-(sqrt(d))^2 or b+c-d+2sqrt(bc) ` .

- Then multiply both denominator and numerator by `(b+c-d)-2sqrt(bc)` , the denominator becomes `(b+c-d)^2-4bc` , which is a rational quantity.

For example,`sqrt(a)or sqrt(b).`

`Consider,`

`12/(3+sqrt(5)- 2sqrt(2))`

`"Step 1: Multiply numerator and denominator by" 3+sqrt(5)-2sqrt(2)`

`= (12(3+sqrt(5)+2sqrt(2)))/((3+sqrt(5))^2-(2sqrt(2))^2)`

`=(12(3+sqrt(5)+2sqrt(2)))/(6+6sqrt(5))`

`"Step 2: To simplify further , multiply the numerator and denominator by" sqrt(5)-1`

`=(2(3+sqrt(5)+2sqrt(2))(sqrt(5)-1))/((sqrt(5)+1)(sqrt(5)-1))`

`=(2+2sqrt(5)+2sqrt(10)-2sqrt(2))/2`

`=1+sqrt(5)+sqrt(10)-sqrt(2) `

**Ex 1`"Find the factor which will rationalise"` `sqrt(3)+root(3)(5)`**

`Sol: Let x=3^(1/2), y=5^(1/3); "then x^6 and y^6 is rational and"`

`x^6-y^6 = (x+y)(x^5-x^4y+x^3y^2-x^2y^3+xy^4-y^5);`

`"Substituting for x and y, the required factor is"`

`3^(5/2)-3^(4/2).5^(1/3)+3^(3/2).5^(2/3)-3^(2/2).5^(3/3)+3^(1/2).5^(4/3)-5^(5/3)`

`or,`

`3^(5/2)-9.5^(1/3)+3^(3/2).5^(2/3)-15+3^(1/2).5^(4/3)-5^(5/3)`

`"the rational product is, "`

`3^(6/2)-5^(6/3)= 3^3-5^2 = 2`

`"Ex 2"``"Express with rational denominator"`

`4/(root(3)(9) -root3(3)+1)`

`"Sol : The expression"` = `(4(3^(1/3)+1))/((3^(1/3)+1)(3^(2/3)-3^(1/3)+1))`

`= (4(3^(1/3)+1))/(3+1) = 3^(1/3)+1`

Pro 1: Express`(5^(1/2)+9^(1/8))+(5^(1/2)-9^(1/8)) "as an equivalent fraction with a rational denominator"`

`"Pro 2 :Find the cube root of" 72-32sqrt(5)`

`Pro 3: "Given" sqrt(5)=2.23607. "find the value of" sqrt(3-sqrt(5))/(sqrt(2)+sqrt(7-3sqrt(5)))`

`Ans1 : (14+5^(3/2).3^(1/4)+5.3^(1/2)+5^(1/2).3(3/4))/11`

`Ans2:" 3-sqrt(5)`

`Ans 3: 0.44721`