√2, √3, √5, √7, √11, ³√5, √8, √50 etc.
√a × √a = a
Example: √5 × √5 = 5
√a × √b = √(ab)
Example: √3 × √12 = √36 = 6
√a ÷ √b = √(a/b)
Example: √8 ÷ √2 = √4 = 2
a√b + c√b = (a + c)√b
Example: 3√5 + 7√5 = 10√5
5. Rationalizing the Denominator
Multiply top & bottom by a suitable surd to remove roots in denominator.
Example:
1/3 =1/3 × 3/3 = 3/3
⭐ 5 Solved Examples of Surds (With Explanation)
Break 27 into perfect square × other number:
27 = 9 × 3
√27 = √(9×3) = √9 × √3 = 3√3
Multiply roots:
√12 × √3 = √(12×3) = √36 = 6
5√2 + 3√2
Same surd → add coefficients:
(5 + 3)√2 = 8√2
1/√5
Multiply by √5/√5:
1/√5 ×√5/√5 = √5/√5
√75 −√12
Break into perfect squares:
√75 = √(25×3) = 5√3
√12 = √(4×3) = 2√3
Now subtract:
5√3 – 2√3 = 3√3
📝Explanation: √49 = 7 because 49 is a perfect square (7×7).
📝 Description: √7 is irrational. Others are pure integers, not surds.
📝 Description:
√9 = 3
√4 = 2
3 × 2 = 6
📝 Description:
18 = 9×2
√18 = 3√2
Oops → Option listing mismatch
Correct: 3√2 → Option A.
📝 Description:
50 = 25×2
√50 = 5√2
📝 Description:
2√7 + 5√7 = (2+5)√7 = 7√7
📝 Description:
√27 = 3√3
So: √3 × 3√3 = 3×3 = 9
Correct: 9 → Option A.
📝 Description: √2/2
📝 Description: √75 ÷ √3 = √(75/3) = √25 = 5
📝 Description:
√8 = √(4×2) = 2√2
2√2 + √2 = 3√2
📝 Description:
√20 = 2√5
2√5 – √5 = √5
Oops correction:
2√5 – √5 = 1√5 → Option C.
📝 Description:
45 = 9×5
√45 = 3√5
📝 Description: 32 = 16×2 → √32 = 4√2
📝 Description: Rule: √a × √b = √(ab)
📝 Description:
Multiply by √5/√5:
(2√5)/5
📝 Description:
√12 = 2√3
3×2√3 = 6√3
📝 Description:
√48 = 4√3
√27 = 3√3
4√3 - 3√3 = √3
📝 Description:
√125 = 5√5
√5 × 5√5 = 5×5 = 25
📝 Description: 7√3 ÷ √3 = 7
📝 Description:
√12 = 2√3
So 2√12 = 4√3
Numerator:
3√3 + 4√3 = 7√3
Now divide:
7√3 ÷ √3 = 7
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