**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.

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.

**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}

** = **48^{2 (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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