Simplifying Radical Expressions :
In this section, you will learn how to simplify radical expressions.
The following steps will be useful to simplify any radical expressions.
(i) Decompose the number inside the radical into prime factors.
(ii) If you have square root (√), you have to take one term out of the square root for every two same terms multiplied inside the radical.
(iii) If you have cube root (3√), you have to take one term out of cube root for every three same terms multiplied inside the radical.
(iv) If you have fourth root (4√), you have to take one term out of fourth root for every four same terms multiplied inside the radical.
(v) Combine the radical terms using mathematical operations.
Example :
√18 + √8 = √(3 ⋅ 3 ⋅ 2) + √(2 ⋅ 2 ⋅ 2)
√18 + √8 = 3√2 + 2√2
√18 + √8 = 5√2
Example 1 :
Simplify the radical expression :
√169 + √121
Solution :
Decompose 169 and 121 into prime factors using synthetic division.
√169 = √(13 ⋅ 13) √169 = 13 |
√121 = √(11 ⋅ 11) √121 = 11 |
So, we have
√169 + √121 = 13 + 11
√169 + √121 = 24
Example 2 :
Simplify the radical expression :
√20 + √320
Solution :
Decompose 20 and 320 into prime factors using synthetic division.
√20 = √(2 ⋅ 2 ⋅ 5) √20 = 2√5 |
√320 = √(2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 5) √320 = 2 ⋅ 2 ⋅ 2 ⋅ √5 √320 = 8√5 |
So, we have
√20 + √320 = 2√5 + 8√5
√20 + √320 = 10√5
Example 3 :
Simplify the radical expression :
√117 - √52
Solution :
Decompose 117 and 52 into prime factors using synthetic division.
√117 = √(3 ⋅ 3 ⋅ 13) √117 = 3√13 |
√52 = √(2 ⋅ 2 ⋅ 13) √52 = 2√13 |
So, we have
√117 - √52 = 3√13 - 2√13
√117 + √52 = √13
Example 4 :
Simplify the radical expression :
√243 - 5√12 + √27
Solution :
Decompose 243, 12 and 27 into prime factors using synthetic division.
√243 = √(3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3) = 9√3
√12 = √(2 ⋅ 2 ⋅ 3) = 2√3
√27 = √(3 ⋅ 3 ⋅ 3) = 3√3
So, we have
√243 - 5√12 + √27 = 9√3 - 5(2√3) + 3√3
Simplify.
√243 - 5√12 + √27 = 9√3 - 10√3 + 3√3
√243 - 5√12 + √27 = 2√3
Example 5 :
Simplify the radical expression :
-√147 - √243
Solution :
Decompose 147 and 243 into prime factors using synthetic division.
√147 = √(7 ⋅ 7 ⋅ 3) = 7√3
√243 = √(3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3) = 9√3
So, we have
-√147 - √243 = -7√3 - 9√3
-√147 - √243 = -16√3
Example 6 :
Simplify the radical expression :
(√13)(√26)
Solution :
Decompose 13 and 26 into prime factors.
13 is a prime number. So, it can't be decomposed anymore.
√26 = √(2 ⋅ 13) = √2 ⋅ √13
So, we have
(√13)(√26) = (√13)(√2 ⋅ √13)
(√13)(√26) = (√13 ⋅ √13)√2
(√13)(√26) = 13√2
Example 7 :
Simplify the radical expression :
(3√14)(√35)
Solution :
Decompose 14 and 35 into prime factors.
√14 = √(2 ⋅ 7) = √2 ⋅ √7
√35 = √(5 ⋅ 7) = √5 ⋅ √7
So, we have
(3√14)(√35) = 3( √2 ⋅ √7)(√5 ⋅ √7)
(3√14)(√35) = 3(√7 ⋅ √7)(√2 ⋅ √5)
(3√14)(√35) = 3(7)√(2 ⋅ 5)
(3√14)(√35) = 21√10
Example 8 :
Simplify the radical expression :
(8√117) ÷ (2√52)
Solution :
Decompose 117 and 52 into prime factors using synthetic division.
√117 = √(3 ⋅ 3 ⋅ 13) √117 = 3√13 |
√52 = √(2 ⋅ 2 ⋅ 13) √52 = 2√13 |
(8√117) ÷ (2√52) = 8(3√13) ÷ 2(2√13)
(8√117) ÷ (2√52) = 24√13 ÷ 4√13
(8√117) ÷ (2√52) = 24√13 / 4√13
(8√117) ÷ (2√52) = 6
Example 9 :
Simplify the radical expression :
(8√3)2
Solution :
(8√3)2 = 8√3 ⋅ 8√3
(8√3)2 = (8 ⋅ 8)(√3 ⋅ √3)
(8√3)2 = (64)(3)
(8√3)2 = 192
Example 10 :
Simplify the radical expression :
(√2)3 + √8
Solution :
(√2)3 + √8 = (√2 ⋅ √2 ⋅ √2) + √(2⋅ 2 ⋅ 2)
(√2)3 + √8 = (2 ⋅ √2) + 2√2
(√2)3 + √8 = 2√2 + 2√2
(√2)3 + √8 = 4√2
Example 11 :
Simplify :
4√(x4/16)
Solution :
4√(x4/16) = 4√(x4) / 4√16
4√(x4/16) = 4√(x ⋅ x ⋅ x ⋅ x) / 4√(2 ⋅ 2 ⋅ 2 ⋅ 2)
4√(x4/16) = x / 2
Example 12 :
Simplify :
3√(125p6q3)
Solution :
3√(125p6q3) = 3√(5 ⋅ 5 ⋅ 5 ⋅ p2 ⋅ p2 ⋅ p2 ⋅ q ⋅ q ⋅ q)
3√(125p6q3) = 5p2q
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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