To evaluate the expression \(5 - 5 \cdot (5 - \sqrt{3})^{12}\) and see if it equals 15, we can simplify it step by step.
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Calculate \(5 - \sqrt{3}\):
- This is a numerical value, approximately \(5 - 1.732 = 3.268\).
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Raise \(3.268\) to the 12th power, which is a very large number:
- However, we will keep it as \((5 - \sqrt{3})^{12}\) for now.
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Multiply by 5:
- This results in \(5 \cdot (5 - \sqrt{3})^{12}\).
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Subtract from 5: \[ 5 - 5 \cdot (5 - \sqrt{3})^{12} \]
Since \((5 - \sqrt{3})^{12}\) is a positive number (as both \(5\) and \(\sqrt{3}\) are positive), multiplying by \(5\) gives us a very large positive number. Thus, \(5 - 5 \cdot (5 - \sqrt{3})^{12}\) will yield a negative number.
Now we can conclude that \(5 - 5 \cdot (5 - \sqrt{3})^{12} \neq 15\).
Therefore, the answer is 2 (false).