Exact-value synthesis
Objective. Combine fractions, surds and π without premature rounding.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Exact-value synthesis combines fractions, surds and π through a calculation while keeping every step in exact form, avoiding decimal rounding until (if ever) a final approximation is requested. This requires adding fractions over a common denominator, simplifying surds by factoring out square numbers, and leaving π as a symbol rather than substituting 3.14. Multiplying or dividing surds uses √a×√b=√(ab), and terms can only be combined if they share the same surd or are both rational. For example, 2π×√8 simplifies to 2π×2√2 = 4π√2, kept exact rather than evaluated to a decimal.
Worked example
Find the exact area of a circle with radius (2+√3) cm, giving your answer in terms of π.
- 1Area = πr², so square the radius: (2+√3)² = 4+4√3+3
- 2Simplify: 4+4√3+3 = 7+4√3
- 3Multiply by π: Area = π(7+4√3)
- 4Write in exact form without converting π or √3 to decimals
Answer: (7+4√3)π cm²
Concept mastery
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Practice
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Start practisingCommon mistake. A common error is rounding an intermediate surd or π value to a decimal partway through a multi-step calculation, which introduces rounding error; always keep √ and π symbols throughout and only convert to a decimal at the very end if asked.
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