Evaluate (-1)^(3/4)

Problem

(−1)(3/4)

Solution

1. Rewrite the expression using Euler's formula, expressing −1 in polar form as e(i*π)

(−1)(3/4)=(e(i*π))(3/4)

2. Apply the power rule for exponents by multiplying the exponents together.

(e(i*π))(3/4)=e(i(3*π)/4)

3. Convert the result back to rectangular form using the identity e(i*θ)=cos(θ)+i*sin(θ)

e(i(3*π)/4)=cos((3*π)/4)+i*sin((3*π)/4)

4. Evaluate the trigonometric functions at the angle (3*π)/4 radians.

cos((3*π)/4)=−√(,2)/2

sin((3*π)/4)=√(,2)/2

5. Combine the real and imaginary parts to find the principal value.

−√(,2)/2+i√(,2)/2

Final Answer

(−1)(3/4)=−√(,2)/2+√(,2)/2*i

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