CUBE ROOTS AND RADICALS

About "Cube roots and radicals"

Cube roots and radicals :

Cube root is inverse operation in finding cubes.

The radical of a number is the value such that, when a number multiplied by itself, for example

   3 x 3 = 9

It is written with a square root symbol " √ " and the number or expression inside the square root symbol is called the radicand.

Finding cube roots and radicals 

To obtain cube root of a number, we can use the prime factorization method.

Step 1 :

Resolve the given number into prime factors.

Step 2 :

Write these factors in triplets such that all three factors in each triplet are equal.

Step 3 :

From the product of all factors, take one from each triplet that gives the cube root of a number.

Operations with radicals :

Addition, subtraction, multiplication and division of radical terms can be performed by some laws. Let us see the rules one by one.

Rule 1 :

Whenever we have two or more root terms which are multiplied with same index, then we can put only one radical and multiply the terms inside the root terms. 

Rule 2 :

Whenever we have two or more root terms which are dividing with same index, then we can put only one root  and divide the terms inside the root sign.

Rule 3 :

nth root of a can be written as a to the power 1/n. Whenever we have power to the power, we can multiply both powers.

Cube roots and radicals - Examples

Example 1 :

Find the cube root of 512

Solution :

Hence cube-root of 512 is 8.

Example 2 :

Find the cube-root of 27 x 64

Solution :

  =  ∛27 x 64 

We can write 27 as 3 x 3 x  3, like wise 64 as 4 x 4 x 4.

  =  ∛3 x 3 x 3 x 4 x 4 x 4

  =  3 x 4

  =  12

Hence the answer is 12.

Example 3 :

Find the cube-root of 125/216

Solution :

Here we need to find the cube-root for a fraction. For that, split the numerator and denominator as much as possible.

  =  ∛125/216

125  =  5 x 5 x 5 and 64  =  4 x 4 x 4

  =  ∛(5 x 5 x 5) /(4 x 4 x 4)

Since we have cube-root, we need to take one for each three same terms.

  =  5/4

Hence the cube root of 125/216 is 5/4.

Example 4 :

Find the cube-root of -512/1000

Solution :

Here we need to find the cube-root for a fraction. In the cube-root we have negative sign.

Whenever we have negative sign inside the cube-root, the answer must have negative sign.

  =  ∛512/1000

512  =  8 x 8 x 8 and 1000  =  10 x 10 x 10

  = - ∛(8 x 8 x 8)/(10 x 10 x 10)

Since we have cube-root, we need to take one for each three same terms.

  =  - 8/10

If it is possible, we may simplify

  =  - 4/5

Hence the cube-root of ∛-512/1000 is -4/5.

Example 5 :

Find the cube-root of 0.027

Solution :

Here we need to find the cube-root for a decimal. 

First let us convert the given decimal as fraction. For that, we have to multiply and divide by 1000.

0.027 x  (1000/1000)  =  27/1000

∛0.027  =  ∛27/1000

  =  ∛(3 x 3 x 3)/(10 x 10 x 10)

  =  3/10

Hence the cube-root of ∛0.027 is 3/10.

Example 6 :

Simplify the following radical terms

4√3, 18√2, -3√3, 15√2

Solution :

  =  4√3 + 18√2 - 3√3 + 15√2 

To simplify the above terms, we need to combine the like terms

  =  4√3  - 3√3 + 18√2 + 15√2

  =  (4 - 3) √3 + (18 + 15) √2

  =  1√3 + 33√2

  =  √3 + 33√2

Example 7 :

Simplify the following radical terms

2∛2, 24∛2, - 4∛2

Solution :

  =  2∛2 + 24∛2 - 4∛2

  =  (2 + 24 - 4) ∛2

 =  22 ∛2

Example 8 :

Multiply ∛13 x ∛5 

Solution :

  =  ∛13 x ∛5 

Since the index of both root terms are same, we can write only one root sign and multiply the numbers.

  =  ∛(13 x 5) 

  =  ∛65

Example 9 :

Multiply 15√54 ÷ 3√6

Solution :

  =  15√54 ÷ 3√6

Since the index of both root terms are same, we can write only one root and divide the numbers.

=  (15/3)√(54/6)

=  5√9  ==>  5√(3 x 3)  ==> 5 x 3  ==> 15

Example 10 :

Multiply (48)1/4 ÷ (72)1/8

Solution :

  =  (48)1/4 ÷ (72)1/8

Since the index of the above  root terms are not same, we need to convert the power 1/4 as 1/8.

  =  (48)(1/4) x (2/2) ÷ (72)1/8

  =  (48)(2/8) ÷ (72)1/8

  =  482 (1/8) ÷ (72)1/8

=  [(48 x 48) ÷ (72)]1/8

=  [2304 ÷ 72]1/8

=  (32)1/8

After having gone through the stuff given above, we hope that the students would have understood "Cube roots and radicals". 

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