Simplifying Radicals

Learn about simplifying radicals concept, before that we shall look at into radical. The Radical is defined as the square root of a number. A radical is used to refer the irrational number. This radical expression has been denoted in the root symbol √ . Thus the process to Simplify Radicals involves expressing the numbers in a simpler form or a reduced form. Students can learn to simplify radicals by the solved examples.

Simplifying radicals include the simplification of radicals denominator before performing basic mathematical operation. And given radical should satisfy the conditions and one should remember for positive value of a the value of 1/aⁿ will always be taken as positive. Learn how to simplify radicals in this page and learn to simplify radicals.

Examples :

Example 1: Find the principal cube root of 125.

Solution:

125 = 5³.

Therefore, the cube root of 125 is 5.

Example 2: Find the Square root of 36.

Solution: 6 x 6 = 36

Therefore, the square root of 36 is 6.

Simplify Radicals with Variables

Simplify radicals with variables , The radicals with variables should be solved n a particular order first of all we should consider the root and then we should consider the radicand (i.e. the part inside the root). Solve the variables inside the root and then solve the factors using the respective root. Such that the radicals with variables should are now in simplest form.

Solved Example

Question: Symplify: `sqrt(144*x^2)`
Solution:

`sqrt(144) = 12`

`sqrt(x^2) =x`

Therefore,

`sqrt( 144 * x^2) =12 * x =12x`

Simplifying Radicals with fractions

How to simplify radicals with fractions Radicals with fractions can be solved by determining the root of the numerator and denominator separately. Simplify radicals with fractions is made much easier when we solve them separately i.e. the numerator and denominator separately.

Solved Examples

Question 1: Simplify: `sqrt(6)/6`
Solution:

`sqrt(6)/6 = sqrt(6)/(sqrt(6)*sqrt(6))`

`= 1/sqrt(6)`

Question 2: Simplify: `sqrt(8) /sqrt(32)`
Solution:

`sqrt(8)/ sqrt(32) = (2sqrt(2))/(4sqrt(2))`

`= 2/4`

`=1/2`

Simplifying Radicals – Multiplication

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 $\sqrt{3}$. $\sqrt{8}$
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$\sqrt{2}$

Simplifying Radicals – Division

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 √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:

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

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

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

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

Properties of Radicals

Adding and Subtracting Radicals