To determine if the equation \( 5 - 5 \cdot (5 - \sqrt{3})^{12} = 15 \) is true or false, we can solve the left side.
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Start with \( 5 - 5 \cdot (5 - \sqrt{3})^{12} \).
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Calculate \( 5 - \sqrt{3} \). The value of \( \sqrt{3} \) is approximately \( 1.732 \), so \( 5 - \sqrt{3} \approx 5 - 1.732 \approx 3.268 \).
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Raise this value to the power of 12: \[ (5 - \sqrt{3})^{12} \approx (3.268)^{12} \] This number will be very large since raising a number greater than 1 to a high power results in a very large number.
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Now, multiply by 5 and calculate: \[ 5 \cdot (5 - \sqrt{3})^{12} \text{ will also be a large number, let's call it } L. \]
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Therefore, the left side of the equation simplifies to: \[ 5 - L \] Since \( L \) is very large, \( 5 - L \) will be a very large negative number.
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Now we can see that the left side is \( 5 - L \) and the right side is 15. Thus we have: \[ 5 - L \neq 15 \]
So the equation \( 5 - 5 \cdot (5 - \sqrt{3})^{12} = 15 \) is false.
Final answer: 2 (for false).
2 answers