Let's simplify the left side of the equation step by step:
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Start with the equation: \[ 5 - 5 \cdot (5 - \sqrt{3})^{12} \]
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Focus on simplifying the term \((5 - \sqrt{3})^{12}\). Since this term is raised to the 12th power and is decreasing (as \(5 - \sqrt{3} \approx 3.267\)), it's critical to note that it's a positive number, and multiplying it with \(-5\) will yield a negative result.
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Let's analyze what happens when we calculate \(5 - (5 \cdot (5 - \sqrt{3})^{12})\). Since \( (5 - \sqrt{3})^{12} \) is positive, \(5 \cdot (5 - \sqrt{3})^{12}\) will be significantly larger than 5.
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Therefore, \(5 - 5 \cdot (5 - \sqrt{3})^{12} < 0\).
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Since \(15\) is certainly greater than \(5 - 5 \cdot (5 - \sqrt{3})^{12}\) which resulted in a negative value, the equation is false.
Thus, the equation \(5 - 5 \cdot (5 - \sqrt{3})^{12} = 15\) is false.
So, the answer is 2 (for false).
1 answer