Rules of Radicals

In math, the radical is one of the number representations which indicate the square root or nth root. The symbol for denoting the radical is "√". For example, `sqrt(a)`, `root(n)(a)` are radicals. The term inside the symbol √ is called as radicand and the 'nth root' or 'square root ' is called as index value. Radical is considered as the opposite operation of exponent. For example, 4² is an exponent which is equal to 16. Now, the radical can be written as `sqrt (16)` =` sqrt (4xx4) ` = ±4. It is also possible to write the radical as an exponent by writing the reciprocal of nth root or square root as a power of radicand. Example: `sqrt(4)` = `4^(1/2)`. The radical has some basic rules for simplifying the problems. This article helps to give the detail explanation about the radical rules with some example problems for simplifying.

I like to share this Simplifying Ratios with you all through my article. 

Explanation to rules for radicals:

Rule 1:
          If the radical has a two radicands in multiplication form, then we can distribute common the index value to each radicand and then relate them by multiplication operation.
          For example, `root(n)(ab)` is the radical which has two radicands with a common index value 'n'. So, we can distribute the index value 'n' to each radicand 'a' and 'b' and then write it as `root(n)(a)` `root(n)(b)`.

  `root(n)(ab)`  =  `root(n)(a)` `root(n)(b)`  

Rules 2:
          If the radical has a two radicands in division form then we can distribute the index value to each radicand and then relate them by division operation.
          For example, `root(n)((a/b))` is the radical which has two radicands in division form with a common index value 'n'. So, we can distribute the index value 'n' to each radicand 'a' and 'b' and then write them as `root(n)(a)` /`root(n)(b)`.

  `root(n)((a/b))` =  `(root(n)(a))` / `(root(n)(b))`

Rules 3:
          If suppose the radicand has a power equals to the index value then it results the radicand without any power.
          For example, if we have radical, `root(n)(x^n)`  then we can also write it as `x^(n/n)`. Now, we can cancel the common term 'n'. So, we get the answer as 'x'.

  `root(n)(x^n)`    = x^(n/n) = x

Rule 4:
          If we have a radical `root(n)(-x)` then we can write it as -`root(n)(x)` where 'n' is odd.
By applying the above basic rules, we can easily simplifying the radicals.

Example Problem - Rules for simplifying radicals:

Example: 1

Simplify:`sqrt(50)`

Solution:
Given: `sqrt(50)`
For simplifying the given radical `sqrt(50)` , use the rules of radicals.
Step 1:

First find the factors of the radicand 50.

Factors of 50 are 25 and 2. (50 = 25 x 2)

Now, the radical can be written as `sqrt(50)` = `sqrt(25xx2)`

Step 2:

We need to apply the rule 1 for simplifying the radical.

The rule is, `root(n)(ab)` = `root(n)(a)` `root(n)(b)`

`sqrt(25xx2)` = `sqrt(25)` `sqrt(2)` (Here, we distribute the square root to both radicands 25 and 2)

= `sqrt(5xx5)` `sqrt(2)`

The square root of 25 is 5. Because the square value of 5 is 25.(5 x 5 = 25)

`sqrt(50)` = 5 `sqrt(2)`

Answer: 5`sqrt(2)`
Example: 2

Simplify: `root(n)(125/27)`

Solution:
Given: `root(3)(125/27)`
We need to apply the rules of radicals to solve this problem.

The radial rule is,

`root(n)((a/b))` =  `(root(n)(a))` / `(root(n)(b))`

Now, the given radical `root(3)(125/27)` can be written as,

`root(3)(125/27)` =`(root(3)(125))/(root(3)(27))`

Step 2:

The cubic root of 125 is 5. Because the cubic values of 5 is 125 (5 x 5 x 5 = 125)

`root(3)(125)` = `root(3)(5xx5xx5)`

= 5

The cubic root of 27 is 3. Because the cubic values of 3 is 27(3 x 3 x 3 = 27)

`root(3)(27)` = `root(3)(3xx3xx3)`

= 3

Step 3:

Now, we get the answer as,

`root(3)(125/27)` =`(root(3)(125))/(root(3)(27))`= `5/3`

Answer: `5/3`
Example: 3

Simplify: `root(5)(25^(1/5))`

Solution:
Given: `root(5)(25^(1/5))` 
Step 1:

The rule of radical for simplifying the exponent is,

  `root(n)(x^n)`    = x^(n/n) = x

Step 2:

When we are applying the rule, we get the answer 25.

`root(5)(25^(1/5)`    = 25^(5/5) =25

Answer: 25
Example: 4

Simplify: `root(3)(-64)`

Solution:
Given: `root(3)(-64)`
Step 1:

The index value of the given radical `root(3)(-64)` is odd number 3.

So, we have to take the negative outside and then find the cubic root ot radicand 64.

`root(3)(-64)` = - `root(3)(64)`

= - 4

Answer: -4