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Home » We use the Euclidean algorithm to find the greatest common factor (GCF) of 68 and 102. - Upplift

We use the Euclidean algorithm to find the greatest common factor (GCF) of 68 and 102. - Upplift

📅 February 28, 2026 👤 scraface

Feb 28, 2026

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📌 Solution: Let the populations be $ a $ and $ b $. Given $ a + b = 10 $ and $ a^2 + b^2 = 50 $. First, compute $ ab $ using $ (a + b)^2 = a^2 + 2ab + b^2 $. Substituting, $ 100 = 50 + 2ab $, so $ ab = 25 $. The sum of cubes is $ a^3 + b^3 = (a + b)^3 - 3ab(a + b) = 1000 - 3 \cdot 25 \cdot 10 = 1000 - 750 = oxed{250} $.

📌 Question: A financial model predicts revenue from two products with $ x + y = 4.5 $ and $ x^2 + y^2 = 10.25 $. What is $ 5x + 5y $?

📌 Solution: Given $ x + y = 4.5 $, multiplying by 5 yields $ 5x + 5y = 5(x + y) = 5 \cdot 4.5 = oxed{22.5} $.

📌 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:

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