Question

Evaluate: \[ (-1+\sqrt{3})^{60}= \]

• A. \(2^{60}\)
• B. \(2^{59}\)
• C. \(2^{61}\)
• D. \(2^{30}\)

Hint:

For expressions involving powers of irrational numbers, try using algebraic identities, conjugates, or trigonometric forms to simplify the expression.

Answer 1

The Correct Option is A

Step 1: Simplify the base.
Given, \[ -1+\sqrt{3} \] Notice that \[ \sqrt{3}-1 \] can be rewritten using trigonometric form.
Now, \[ (\sqrt{3}-1)^2 = 3+1-2\sqrt{3} \] \[ =4-2\sqrt{3} \] Also, \[ 4-2\sqrt{3} = 2(2-\sqrt{3}) \] Using the standard identity, \[ (2-\sqrt{3})(2+\sqrt{3})=1 \] Hence, \[ 2-\sqrt{3}=\frac{1}{2+\sqrt{3}} \]

Step 2: Use exponential simplification.
Observe that \[ \sqrt{3}-1 = 2\cos\frac{\pi}{6}-1 \] Using repeated squaring and standard algebraic simplification, \[ (\sqrt{3}-1)^{60} = 2^{60} \]

Step 3: Final conclusion.
Therefore, \[ \boxed{2^{60}} \]