Rules of Simplifying Radicals

Introduction to rules of simplifying radicals:

Radical symbol used to indicate the square root or nth root. Radical is an algebraic group, a concept in algebraic group theory. A branch of mathematics, a radical of a ring is an ideal of elements of the ring. Radical of a module, in the theory of modules, the radical of a module is a component in the theory of structure and classification. The radical sign is v .                 (source: wikipedia)

Rules of Simplifying Radicals:

Radical is a perfect and significant idea in abstract algebra. The cubic root of x know how to be expressed as   `root(5)(x)`

`root(n)(ab)`  =  `root(n)(a)` `root(n)(b)` 
`root(n)((a/b))` =  `(root(n)(a))` / `(root(n)(b))`
`root(n)(x)`  = (x)(1/n)    and      `root(n)(x^n)`  = x(n/n) = x
`root(n)(a)`m =( `root(n)(a)` )m = (a1/n )m = am/n
` sqrt(-1)` ×  `sqrt(-1)` = -1 where as  `sqrt((-1)*(-1))` = 1


Examples for Rules of Simplifying Radicals:

Example 1:

Simplifying the rules of given radical expression:    `sqrt((9x^2)/(y^5z^5)) `

Solution:

Step 1: Multiplying and divide by  yz

Step 2:      ` sqrt((9x^2)/(y^5z^5))` = `sqrt((9x^2 * yz) / (y^5z^5 *yz))`

Step 3: Multiply the variable with exponent

= `sqrt((9x^2yz) / (y^6z^6))`

Step 4: Square root of x2y6z6 = xy3z3

Step 5: Square root of 9 = `sqrt(3 * 3) ` = 3

=` ((3x) / (y^3z^3)) sqrt(yz)`

so the answer is  ` ((3x) / (y^3z^3)) sqrt(yz)`

Example 2:

Simplifying the rules of radical expression: `sqrt(b^4/a^7) ` + `sqrt(a)`  = `sqrt(a)`  ` ((b+a^3)/a^3)`

Solution:

Step 1:  Multiplies and divides by  a in first term

Step 2:      `sqrt(b^4/a^7)` = `sqrt ((b^4* a) / (a^7 *a))`

Step 3: Multiply the variable with exponent

= `sqrt ((b^4 * a) / (a^8))`

Step 4:  Square root of b4 a = b2 `sqrt(a)`

Step 5:   Square root of a8 = a4

`sqrt ((b^4 * a) / (a^8))`   = ` b^2/a^4 sqrt(a)`

Step 6:  Adding the both term  `sqrt(b^4/a^7) ` + `sqrt(a)`    =` b^2/a^4 sqrt(a)`  +  `sqrt(a)`

= `sqrt(a)` ( `b^2/a^4 ` +1 )

Step 7:         = `sqrt(a)`  ` ((b^2+a^4)/a^4)`

Hence the given radical expression has been proved

Example 3:

Simplifying the rules of  the given radical expression: `sqrt(5x^3) (sqrt(9x^3))`

Solution:

Step 1: the given radical expression is

=   `sqrt(5x^3) (sqrt(9x^3))` 

Step 2:     `sqrt(5x^3) (sqrt(9x^3))`

=` (xsqrt(5x) (3)xsqrtx)`

Step 3:       `3x^2 sqrt(5 * x * x) `                                     

Step 4:       3x2x `sqrt(5)`                                                                 

= 3 x3`sqrt5`  

so the answer is  3 x3`sqrt5`  

Example 4:

Simplifying the rules of the given radical expression :  `(root3 16)^3 * (sqrt49)^2`

Solution:

Step 1: Given radicals ` (root3 16)^3 * (sqrt49)^2`

Step 2:     = `(root3 16)^3 * (sqrt49)^2`

Step 3: Radical rules is ( `root(n)(a)` )m = am/n

Step 4:    = 163/3 * 492/2

161 * 491

Step 5:       16 * 49

Step 6:         so the answer is 784