In this section, you will learn how to simplify radical expressions.
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.
Step 1 :
Decompose the number inside the radical sign into prime factors.
Step 2 :
Take one number out of the radical for every two same numbers multiplied inside the radical sign.
Step 3 :
Simplify.
Examples :
Example 1 :
Simplify :
√20 - √225 + √80
Solution :
Decompose 20, 225 and 80 into prime factors using synthetic division.
√20 = √2 ⋅ 2 ⋅ 5 = 2√5
√225 = √5 ⋅ 5 ⋅ 3 ⋅ 3 = 5 ⋅ 3 = 15
√225 = √2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 5 = (2 ⋅ 2)√5 = 4√5
Then, we have
√20 - √225 + √80 = 2√5 - 15 + 4√5
√20 - √225 + √80 = 6√5 - 15
√20 - √225 + √80 = 6√5 - 15
√20 - √225 + √80 = 3(2√5 - 5)
Example 2 :
Simplify :
√27 + √75 + √108 - √48
Solution :
Decompose 27, 75, 48 and 108 into prime factors using synthetic division.
√27 = √(3 ⋅ 3 ⋅ 3) = 3√3
√75 = √(5 ⋅ 5 ⋅ 3) = 5√3
√108 = √(3 ⋅ 3 ⋅ 3 ⋅ 2 ⋅ 2) = 3 ⋅ 2 ⋅ √3 = 6√3
√48 = √(2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3) = 2 ⋅ 2 ⋅ √3 = 4√3
Then, we have
√27 + √75 + √108 - √48 = 3√3 + 5√3 + 6√3 - 4√3
√27 + √75 + √108 - √48 = 10√3
Example 3 :
Simplify the following radical expression
5√28 - √28 + 8√28
Solution :
5√28 - √28 + 8 √28
Because all the terms in the above radical expression are like terms, we can simplify as given below.
5√28 - √28 + 8√28 = 12√28
5√28 - √28 + 8√28 = 12√(2 ⋅ 2 ⋅ 7)
5√28 - √28 + 8√28 = 12 ⋅ 2√7
5√28 - √28 + 8√28 = 24√7
Example 4 :
Simplify the following radical expression
9√11 - 6√11
Solution :
9√11 - 6√11
Because the terms in the above radical expression are like terms, we can simplify as given below.
9√11 - 6√11 = 3√11
Example 5 :
Simplify the following radical expression
7√8 - 6√12 - 5√32
Solution :
7√8 - 6√12 - 5√32
Decompose 8, 12 and 32 into prime factors.
7√8 = 7√(2 ⋅ 2 ⋅ 2) = 7 ⋅ 2√2 = 14√2
6√12 = 6√(2 ⋅ 2 ⋅ 3) = 6 ⋅ 2√3 = 12√3
5√32 = √(2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2) = 5 ⋅ 2 ⋅ 2 ⋅ √2 = 20√2
Then, we have
7√8 - 6√12 + 5√32 = 14√2 - 12√3 - 20√2
7√8 - 6√12 + 5√32 = 14√2 - 12√3 - 20√2
7√8 - 6√12 + 5√32 = -6√2 - 12√3
7√8 - 6√12 + 5√32 = -6(√2 + 2√3)
Example 6 :
Simplify the following radical expression
2√99 + 2√27 - 4√176 - 3√12
Solution :
Decompose 99, 27, 176 and 12 into prime factors.
2√99 = 2√(3 ⋅ 3 ⋅ 11) = 2 ⋅ 3√11 = 6√11
2√27 = 2√(3 ⋅ 3 ⋅ 3) = 2 ⋅ 3√3 = 6√3
4√176 = √(2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 11) = 4 ⋅ 2 ⋅ 2√11 = 16√11
3√12 = 3√(2 ⋅ 2 ⋅ 3) = 3 ⋅ 2√3 = 6√3
Then, we have
2√99 + 2√27 - 4√176 - 3√12 = 6√11 + 6√3 - 16√11 - 6√3
2√99 + 2√27 - 4√176 - 3√12 = -10√11
After having gone through the stuff given above, we hope that the students would have understood how to simplify radical expressions.
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