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Simplifying Radicals

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

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Simplify Radicals with Variables

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To solve a radical with variable. follow the given steps.

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 $\sqrt{144 x^2}$
$\sqrt{144}$ = 12

$\sqrt{x^2}$ =x


$\sqrt{144  x^2}$ =12 $\times x$ =12x

Simplifying Radicals with fractions

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


Question 2: Simplify: $\frac{\sqrt{8}}{\sqrt{32}}$
$\frac{\sqrt{8}}{\sqrt{32}}$ = $\frac{2\sqrt{2}}{4\sqrt{2}}$

= $\frac{2}{4}$


Simplifying Radicals Multiplication

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When multiplying radicals, radicals of same radical powers can be brought under a single radical sign, the terms inside can be multiplied and simplified.

Solved Examples

Question 1: Simplify $\sqrt{3}$. $\sqrt{8}$
$\sqrt{3}$. $\sqrt{8}$ = $\sqrt{24}$ = $\sqrt{4}$ $\sqrt{6}$ = 2$\sqrt{6}$

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

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

Simplifying Radicals Division

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

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


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

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