To add and subtract radicals, we need to be aware of like 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.
The following steps will be useful to simply radical expressions
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, if the radical is a square root.
Take one number out of the radical for every three same numbers multiplied inside the radical sign, if the radical is a cube root.
Step 3 :
Simplify.
Examples :
√4 = √(2 ⋅ 2) = 2
√16 = √(2 ⋅ 2 ⋅ 2 ⋅ 2) = 2 ⋅ 2 = 2
3√27 = 3√(3 ⋅ 3 ⋅ 3) = 3
3√125 = 3√(5 ⋅ 5 ⋅ 5) = 5
Simplify the following radical expressions :
Example 1 :
√3 + √12
Solution :
= √3 + √12
= √3 + √(2 ⋅ 2 ⋅ 3)
= √3 + 2√3
= 3√3
Example 2 :
√75 + √25
Solution :
= √75 + √3
= √(5 ⋅ 5 ⋅ 3) + √3
= 5√3 + √3
= 6√3
Example 3 :
√18 + √98
Solution :
= √18 + √98
= √(3 ⋅ 3 ⋅ 2) + √(7 ⋅ 7 ⋅ 2)
= 3√2 + 7√2
= 10√2
Example 4 :
√5 + 2√5 - 5√5
Solution :
= √5 + √20 - √125
= √5 + √(2 ⋅ 2 ⋅ 5) - √(5 ⋅ 5 ⋅ 5)
= √5 + 2√5 - 5√5
= -2√5
Example 5 :
√5 + 3√7 - 4√5 - 5√7
Solution :
√5 + 3√7 - 4√5 - 5√7
Group the like radicals.
= (√5 - 4√5) + (3√7 - 5√7)
Combine the like radicals.
= (-3√5) + (-2√7)
= -3√5 - 2√7
Example 6 :
3√3 + 4√3 - √2
Solution :
= 3√3 + 4√3 - √2
Group the like radicals.
= (3√3 + 4√3) - √2
Combine the like radicals.
= 7√3 - √2
Example 7 :
2(√5 - √3) + 3(√3 - √5)
Solution :
= 2(√5 - √3) + 3(√3 - √5)
Use Distributive Property.
= 2√5 - 2√3 + 3√3 - 3√5
Group the like radicals.
= (2√5 - 3√5) + (-2√3 + 3√3)
Combine the like radicals.
= -√5 + √3
Example 8 :
√8 + √18
Solution :
= √8 + √18
= √(2 ⋅ 2 ⋅ 2) + √(3 ⋅ 3 ⋅ 2)
= 2√2 + 3√2
= 5√2
Example 9 :
3√16 + 3√54
Solution :
= 3√16 + 3√54
= 3√(2 ⋅ 2 ⋅ 2 ⋅ 2) + 3√(3 ⋅ 3 ⋅ 3 ⋅ 2)
= 23√2 + 33√2
= 53√2
Example 10 :
√25 + 53√64
Solution :
= √25 + 53√64
= √(5 ⋅ 5) + 53√(4 ⋅ 4 ⋅ 4)
= 5 + 5(4)
= 5 + 20
= 25
Example 11 :
√12w + √27w
Solution :
= √(12w) + √(27w)
= √(2 ⋅ 2 ⋅ 3 ⋅ w) + √(3 ⋅ 3 ⋅ 3 ⋅ w)
= 2√3w + 3√3w
= 5√3w
Example 12 :
√45y3 + √25y3
Solution :
= √45y3 + √25y3
= √(3 ⋅ 3 ⋅ 5 ⋅ y ⋅ y ⋅ y) + √(2 ⋅ 2 ⋅ 5 ⋅ y ⋅ y ⋅ y)
= 3y√5y + 2y√5y
=5y√5y
= 5y√5y
Example 13 :
3√8x3y6 + √9x2y4
Solution :
= 3√8x3y6 + √9x2y4
= 3√(2 ⋅ 2 ⋅ 2 ⋅ x ⋅ x ⋅ x ⋅ y2 ⋅ y2 ⋅ y2) + √(3 ⋅ 3 ⋅ x ⋅ x ⋅ y2 ⋅ y2)
= 2xy2 + 3xy2
= 5xy2
Example 14 :
√4p2q4 - 3√125p3q6
Solution :
= √4p2q4 - 3√125p3q6
= √(2 ⋅ 2 ⋅ p ⋅ p ⋅ q2 ⋅ q2) - √(5 ⋅ 5 ⋅ 5 ⋅ p ⋅ p ⋅ p ⋅ q2 ⋅ q2 ⋅ q2)
= 2pq2 - 5pq2
= - 3pq2
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