Evaluate (-256)^(3/4)

Problem

(−256)(3/4)

Solution

1. Rewrite the expression using radical notation to separate the power and the root.

(−256)(3/4)=(√(4,−256))3

2. Identify the nature of the fourth root of a negative number.

√(4,−256)

3. Determine if a real solution exists. Since the index of the root is even (4) and the radicand is negative (-256), there is no real number that, when multiplied by itself four times, results in a negative value.

√(4,−256)∉ℝ

4. Apply complex number rules if required. In the complex plane, the principal fourth root is found using the formula r(1/n)*e(i*θ/n) For −256 the magnitude r=256 and the argument θ=π

√(4,−256)=256(1/4)*e(i*π/4)

5. Simplify the fourth root of the magnitude.

256(1/4)=4

6. Calculate the principal value by raising the result to the third power.

(4*e(i*π/4))3=4*e(i*3*π/4)

7. Convert back to rectangular form using Euler's formula e(i*x)=cos(x)+i*sin(x)

64*(cos(3*π/4)+i*sin(3*π/4))

8. Substitute the trigonometric values.

64*(−√(,2)/2+i√(,2)/2)

9. Distribute the constant.

−32√(,2)+32*i√(,2)

Final Answer

(−256)(3/4)=−32√(,2)+32*i√(,2)

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