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Home » 5Question: Rationalize the denominator of $\displaystyle rac3{\sqrt7 - \sqrt2}$ and simplify. - Midis

5Question: Rationalize the denominator of $\displaystyle rac3{\sqrt7 - \sqrt2}$ and simplify. - Midis

📅 March 12, 2026 👤 scraface

Mar 12, 2026

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📌 \langle 1, 0, 4 \rangle \times \langle 2, -1, 3 \rangle = \langle (0)(3) - (4)(-1), -[(1)(3) - (4)(2)], (1)(-1) - (0)(2) \rangle = \langle 0 + 4, -(3 - 8), -1 - 0 \rangle = \langle 4, 5, -1 \rangle

📌 Thus, the general solution is a line, but a particular solution is:

📌 \boxed{\langle 1, 0, 4 \rangle}

📌 This vector models a rotational forcing compatible with nutrient gradient symmetry in the engineered extremophile model.

📌 Question: Find the minimum value of $ (\cos x + \sec x)^2 + (\sin x + \csc x)^2 $ over $ x \in (0, \frac{\pi}{2}) $, arising in optimization of energy efficiency in a synthetic metabolic loop.

📌 Expand both terms:

📌 = \cos^2 x + 2\cos x \sec x + \sec^2 x + \sin^2 x + 2\sin x \csc x + \csc^2 x

📌 Since $ \cos x \sec x = 1 $, $ \sin x \csc x = 1 $, and $ \cos^2 x + \sin^2 x = 1 $, we get:

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