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Surds
Chapter summary, hard words and model exam answers for ICSE Class 10 Hindi.
Free online summary and notes (ICSE Class 10 Hindi). Read it here, no PDF download needed.
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Mathematics · CBSE 10 · ICSE 10 · GCSE (AQA, Edexcel, OCR)
Summary
√4 = 2 exactly, so it is not a surd. √2 = 1.41421… runs forever, so it is irrational - a surd. We keep it as √2 to stay exact instead of rounding.
√12 = √(4 × 3) = √4 × √3 = 2√3. Always pick the biggest square factor so one step is enough: √75 = √(25 × 3) = 5√3.
3√2 + 5√2 = 8√2, exactly like 3x + 5x = 8x. Simplify first if needed: √8 + √18 = 2√2 + 3√2 = 5√2. But √2 + √3 cannot be combined.
For 1/√3, multiply by √3/√3 to get √3/3. For (2 + √3) on the bottom, multiply by the conjugate (2 − √3): the difference of squares (2 + √3)(2 − √3) = 4 − 3 = 1 leaves a whole-number denominator.
Hard words & meanings
| surd | an irrational root that cannot be simplified to a rational number |
| irrational number | a real number that cannot be written as a fraction p/q of integers |
| radicand | the number or expression under the root sign |
| perfect square | an integer that is the square of an integer, e.g. 1, 4, 9, 16, 25 |
| like surds | surds with the same radicand, e.g. 3√5 and 7√5 |
| conjugate | the partner of (a + √b) is (a − √b); their product removes the surd |
| rationalise | to rewrite a fraction so no surd appears in the denominator |
| difference of two squares | the identity (a + b)(a − b) = a² − b², used to make surds rational |
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