Math Radicals Practice

Radical symbol used to indicate the square root or nth root. Radical of an algebraic group, a concept in algebraic group theory. Radical of a ring, in ring theory, a branch of mathematics, a radical of a ring is an ideal of "bad" 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. Radical of an ideal, an important concept in abstract algebra. The radical symbol is ' √ ' . The cubic root of x can be expressed as `root(3)(x)`

Understanding Addition Practice is always challenging for me but thanks to all math help websites to help me out. 
                                                                                                                                                                   Source Wikipedia.

Basic identities and properties of radical:

         Multiplication property rule     `root(n)(x)` *`root(n)(y)`  = `root(n)(x*y)`
         Division property rule `sqrt(x/y)`  =  `sqrt x /sqrt y`
       General Relation between exponential and radical expression     `root(n)(x)`^m =( `root(n)(x)` )^m = (x^1/n )^m = x^m/n

√ -1 ×√-1 = -1 where as  `sqrt((-1) * (-1)) ` = 1

Math radicals practice problems:

Math radicals practice problem 1:

     Simplify the radical expression:` (sqrt(16a^4)) (sqrt((-25)a^3))`
    Solution:
          ` (sqrt(16a^4)) (sqrt((-25)a^3))` = ` (sqrt(16a^4)) (sqrt((-25)a^3))`
                                                    = (4a²   a `sqrt(-25a)` )
                                                    = 4a³ `sqrt((-1)25a)`                     we know,   √(-1) = i
                                                    = i 4a³ `sqrt(25a)`
                                                    = i 4a³ (5) `sqrt(a)`
                                                    =    i 20a³ `sqrt(a)`
        Answer:      i 20a³ `sqrt(a)`
Math radicals practice problem 2:
      Simplify the radicals:    `sqrt((8x^2)/(y^5z^7)) `
     Solution:
          Step 1: Multiply and divide by  yz
                    ` sqrt((8x^2)/(y^5z^7))` = `sqrt((8x^2 * yz) / (y^5z^7 *yz))`
          Step 2: Multiply the variable with exponent
                                       = `sqrt((8x^2yz) / (y^6z^8))`
          Step 3: Square root of x²y⁶z⁸ = xy³z⁴
                      Square root of 8 = 2`sqrt2 `   
                           So the answer = ` ((2x) / (y^3z^4)) sqrt(2yz)`
      Answer:   ` ((2x) / (y^3z^4)) sqrt(2yz)`

Math radicals practice problem 3:
           Simplify the radicals : `sqrt(64)/sqrt49 + sqrt169 `
Solution:
             Step 1: Very smallest factors of 64 = 8 × 8
                             `sqrt(64)` = `sqrt(8 * 8) ` = 8
             Step 2: Very smallest factors of 169 = 13 × 13
                              `sqrt(169)` = `sqrt(13 * 13)`
             Step 3: Take square root of 169 = 13
             Step 4: Very smallest factors of 49 = 7 × 7
                          `sqrt(49)` = `sqrt(7 * 7)`
             Step 5:  Take square root of 49= 7
             Step 6: so,  `sqrt(64) /sqrt(49)` = `8 / 7 `
             Step 7: Now ,  `sqrt(64)/sqrt49 + sqrt169 ` =  `8 / 7 `+ 13 = `(8 + 91)/7`
             Step 8:  Simplification of  `sqrt(64)/sqrt49 + sqrt169 ` = ` 99/7 `
    Answer:   ` 99/7 `

Math radicals practice problem 4:
       Simplify the radicals:  `(sqrt36) (3sqrt(45a^2))`
    Solution:
             `(sqrt36) (3sqrt(45a^2))` = (`sqrt(6 xx 6)` ) (`3sqrt(45a^2)` )
                                          = `(6) ( 3a sqrt(9 * 5))`
                                          = (6) (3a(`3sqrt5` ))
                                          = 6 (9a) `sqrt5`
                                          = 54a `sqrt5`
    Answer:  54a `sqrt5`

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Math radicals practice problem 5:
          Find the cubic root of 135
      Solution:
          The very smallest factor of 135  = 3 * 3 * 3 * 5
          Cubic root of 135         `root(3)(135)`  = `root(3)(3 * 3 * 3 * 5)`
                                                                    =  `3 ` `root(3)(5)`

      Answer:  `3 ` `root(3)(5)`