SIMPLIFY RADICAL EXPRESSIONS
Simplify radical expressions :
Before learning how to simplify radical expressions, we must know about like radicals and unlike radicals.
Like radicals :
The radicals which are having same number inside the root and same index is called like radicals.
Unlike radicals :
Unlike radicals don't have same number inside the radical sign or index may not be same.
We can add and subtract like radicals only.
Let us see some example problems to understand this topic
Example 1 :
Simplify the following radical expression
√27 + √75 + √108 - √48
Solution :
= √27 + √75 + √108 - √48
First we have to split the given numbers inside the radical as much as possible.
= √(3 x 3 x 3) + √(3 x 5 x 5) +
√(3 x 3 x 2 x 2 x 2) - √(2 x 2 x 2 x 2 x 2)
= 3 √3 + 5 √3 + 2 x 3 √2 - 2 x 2 √2
= 3 √3 + 5 √3 + 6 √2 - 4 √2
= (3 + 5) √3 + (6-4) √2
= 8 √3 + 2 √2
Hence 8 √3 + 2 √2 is the answer.
Example 2 :
Simplify the following radical expression
5√28 - √28 + 8 √28
Solution :
= 5√28 - √28 + 8 √28
Since they are like radicals, first we can combine them as one term.
= (5 - 1 + 8)√28
= (13 - 1)√28
= 12√28
We can simplify √28 , for that we have to factor 28.
= 12√(2 x 2 x 7)
= (12 x 2 )√7
= 24 √7
Hence 24 √7 is the answer.
Example 3 :
Simplify the following radical expression
9√11 - 6√11
Solution :
= 9√11 - 6√11
Since they are like radicals, first we can combine them as one term.
= (9-6)√11
= 3√11
Since 11 is a prime number, we can not simplify this hereafter.
Hence 3√11 is the answer.
Example 4 :
Simplify the following radical expression
7 √8 - 6 √12 + 5 √32
Solution :
= 7 √8 - 6 √12 - 5 √32
Since they are not like radicals we have to split the numbers inside the radicals as much as possible to make them as like radicals.
= 7 √(2 x 2 x 2) - 6 √(2 x 2 x 3) - 5 √(2 x 2 x 2 x 2 x 2)
= (7 x 2) √2 - (6 x 2) √3 - (5 x 2 x 2) √2
= 14 √2 - 12 √3 - 20 √2
= (14 - 20) √2 - 12 √3
= -6 √2 - 12 √3
Hence -6 √2 - 12 √3 is the answer.
Example 5 :
Simplify the following radical expression
2 √99 + 2 √27 - 4 √176 - 3√12
Solution :
= 2 √99 + 2 √27 - 4 √176 - 3√12
Since they are not like radicals we have to split the numbers inside the radicals as much as possible to make them as like radicals.
= 2 √(3 x 3 x 11) + 2 √(3 x 3 x 3) - 4 √(2 x 2 x 2 x 2 x 11) -
3 √(2 x 2 x 3)
= (2 x 3) √11 + (2 x 3) √3 - (4 x 2 x 2) √11 - (3 x 2) √3
= 6 √11 + 6 √3 - 16 √11 - 6 √3
= 6 √11 - 16 √11 + 6 √3 - 6 √3
= (6 - 16) √11 + (6 - 6) √3
= -10 √11 + 0 √3
= -10 √11
Hence -10 √11 is the answer.
Related pages
- Properties of radicals
- Simplifying radical expressions worksheets
- Square roots
- Ordering square roots from least to greatest
- Operations with radicals
- How to simplify radical expressions
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