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The Radical is defined as root of a number. It can be a square root, cube root or any other root. $\sqrt{3},\sqrt[3]{4},\sqrt[7]{6}$ are all examples of radicals only. Square root is the most commonly used radical. A radical is used to refer the irrational number. This radical expression has been denoted in the root symbol $\sqrt{}$. Thus the process to simplify a radical is to write it in simplest form possible. To simplify a radical the first step is check if the radicand is a multiple of any perfect square, and then to take that perfect square out of the radical. Let us see some examples to understand the radicals and their simplification.

Example 1: Find the principal cube root of 125.

Solution: 125 = $5^3$

Therefore, the cube root of 125 is 5.

Example 2: Simplify $\sqrt{75}$.

Solution: 75 = $5 \times 5\times 3$. Hence, 75 is a multiple of 25 which is  a perfect square. Thus, 25 is taken out of the radical.

Therefore, $\sqrt{75}=5 \sqrt{3}$.

1. Separately take numbers and variables in radicals.
2. Find out if there is a perfect square in the number part. If yes, extract it out of the radical.
3. Keep repeating step 2 till there is no perfect square left in the radicand.
4. Find out if the variable part has any perfect square. If yes, extract it out of the radical.

### Solved Example

Question: Write the simplified form of $\sqrt{144 x^2}$
Solution:

$\sqrt{144}$ = 12

$\sqrt{x^2}$ =x

Therefore,

$\sqrt{144 x^2}$ =12 $\times x$ =12x

To simplify a radical with fractions, the denominator and the numerator are needed to be solved separately. To solve a radical with fractions follow the given steps.
1. Find out if their is a perfect square in the radicand of the denominator and take it out.
2. Repeat the step1 for the numerator.
3. If still a radical is left is the denominator, multiplying both the numerator and the denominator by that radical.
4. Simplify the result.

### Solved Examples

Question 1: Simplify: $\frac{6}{\sqrt{6}}$
Solution:

$\frac{6}{\sqrt{6}}$ = $\frac{6\sqrt{6}}{\sqrt{6}\sqrt{6}}$
= ${\sqrt{6}}$

Question 2: Simplify: $\frac{\sqrt{8}}{\sqrt{32}}$
Solution:

$\frac{\sqrt{8}}{\sqrt{32}}$ = $\frac{2\sqrt{2}}{4\sqrt{2}}$

= $\frac{2}{4}$

$\frac{1}{2}$

### Solved Examples

Question 1: Simplify $\sqrt{3}$. $\sqrt{8}$
Solution:

$\sqrt{3}$. $\sqrt{8}$ = $\sqrt{24}$ = $\sqrt{4}$ $\sqrt{6}$ = 2$\sqrt{6}$

The FOIL method can also be used in multiplication of radical expressions.

Question 2: Evaluate (5 + 2$\sqrt{8}$)(3 - 4$\sqrt{8}$)
Solution:

(5 + 2$\sqrt{8}$)(3 - 4$\sqrt{8}$) = (5)(3) + (5)( - 4$\sqrt{8}$) + 2$\sqrt{8}$(3) + 2$\sqrt{8}$( - 4$\sqrt{8}$)

= 15 -20$\sqrt{8}$ +6$\sqrt{8}$ - 8$\sqrt{8}$$\sqrt{8} = 15 -14\sqrt{8} – 8(8) = 15 -14\sqrt{4}$$\sqrt{2}$ – 64

= 15 – 64 -14(2)$\sqrt{2}$

=– 49 -28$\sqrt{2}$

Here also, radicals of same radical powers can be brought under a single radical sign, the terms inside can be divided and simplified. You may notice that the denominator is multiplied by $\sqrt{3}$ to make it free from radical term. It is a convention that radicals are avoided in the denominators and the process to eliminate radicals in denominators is called ‘rationalizing the denominator’.

### Solved Example

Question: Evaluate $\frac{\sqrt{8}}{\sqrt{6}}$
Solution:

$\frac{\sqrt{8}}{\sqrt{6}}$ = $\sqrt{\frac{8}{6}}$ = $\sqrt{\frac{4}{3}}$ =$\frac{\sqrt{4}}{\sqrt{3}}$

= $\frac{2}{\sqrt{3}}$

$\frac{(2)(\sqrt{3})}{(\sqrt{3})(\sqrt{3})}$

= $\frac{(2\sqrt{3})}{(3)}$

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