Is the equation 5−5⋅(5–√3)12=15true or false? Enter 1 for true or 2 for false.(1 point)

Answer 1

Let's simplify the left side of the equation step by step:

1. Start with the equation: \[ 5 - 5 \cdot (5 - \sqrt{3})^{12} \]

2. 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.

3. 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.

4. Therefore, \(5 - 5 \cdot (5 - \sqrt{3})^{12} < 0\).

5. 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).