Simplify ( square root of 2-i)^4

Problem

(√(,2−i))^4

Solution

1. Rewrite the expression using exponent rules. Since the square root is equivalent to the power of 1/2 the expression becomes ((2−i)^(1/2))^4

2. Multiply the exponents. Using the rule (a^m)^n=a^(m⋅n) we calculate 1/2⋅4=2

3. Simplify the expression to a squared binomial. The expression is now (2−i)^2

4. Expand the binomial using the formula (a−b)^2=a^2−2*a*b+b^2

(2−i)^2=2^2−2*(2)*(i)+i^2

5. Substitute the value of i^2 Since i is the imaginary unit, i^2=−1

4−4*i+(−1)

6. Combine the real parts. Subtract 1 from 4 to get the final simplified form.

3−4*i

Final Answer

(√(,2−i))^4=3−4*i

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