Evaluate (-81)^(3/4)

Problem

(−81)(3/4)

Solution

1. Rewrite the expression using the property of rational exponents a(m/n)=(√(n,a))m

(−81)(3/4)=(√(4,−81))3

2. Identify the nature of the root. The expression involves an even root (index 4) of a negative number.

√(4,−81)

3. Apply the definition of the imaginary unit i=√(,−1) to express the fourth root in terms of complex numbers.

√(4,−81)=√(4,81)⋅√(4,−1)

4. Calculate the fourth root of 81.

√(4,81)=3

5. Determine the principal fourth root of −1 using Euler's formula e(i*θ)=cos(θ)+i*sin(θ) Since −1=e(i*π) the principal root is e(i*π/4)

√(4,−1)=cos(π/4)+i*sin(π/4)

√(4,−1)=√(,2)/2+(i√(,2))/2

6. Combine the results to find the value of the fourth root.

√(4,−81)=3*(√(,2)/2+(i√(,2))/2)

√(4,−81)=(3√(,2))/2+(3*i√(,2))/2

7. Cube the result to find the final value. Using the polar form 3*e(i*π/4)

(3*e(i*π/4))3=3*e(i*3*π/4)

27*(cos((3*π)/4)+i*sin((3*π)/4))

8. Simplify the trigonometric values.

cos((3*π)/4)=−√(,2)/2

sin((3*π)/4)=√(,2)/2

9. Distribute the constant.

27*(−√(,2)/2+(i√(,2))/2)=−(27√(,2))/2+(27*i√(,2))/2

Final Answer

(−81)(3/4)=−(27√(,2))/2+(27*i√(,2))/2

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