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Home » Sum $ \approx 0.1 + 0.0588235 + 0.0384615 + 0.027027 = 0.224312 $ - AMAZONAWS

Sum $ \approx 0.1 + 0.0588235 + 0.0384615 + 0.027027 = 0.224312 $ - AMAZONAWS

📅 March 5, 2026 👤 scraface

Mar 05, 2026

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📌 Question: Rationalize the denominator of $ rac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}} $.

📌 Solution: Multiply numerator and denominator by $ \sqrt{7} + \sqrt{3} $:

📌 rac{(\sqrt{7} + \sqrt{3})^2}{(\sqrt{7})^2 - (\sqrt{3})^2} = rac{7 + 2\sqrt{21} + 3}{7 - 3} = rac{10 + 2\sqrt{21}}{4} = rac{5 + \sqrt{21}}{2}.

📌 The rationalized form is $ oxed{rac{5 + \sqrt{21}}{2}} $.

📌 Solution: Recognize it as a difference of cubes: $ (2a)^3 - (3b)^3 = 8a^3 - 27b^3 $. Alternatively, expand directly:

📌 2a \cdot 4a^2 + 2a \cdot 6ab + 2a \cdot 9b^2 - 3b \cdot 4a^2 - 3b \cdot 6ab - 3b \cdot 9b^2 = 8a^3 + 12a^2b + 18ab^2 - 12a^2b - 18ab^2 - 27b^3.

📌 Combine like terms: $ 8a^3 - 27b^3 $. The expanded form is $ oxed{8a^3 - 27b^3} $.

📌 Question: Two biotechnology strains grow according to $ y = 3x + 2 $ and $ y = -2x + 12 $. Find their intersection point.

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