Evaluate (-16)^(3/4)

Problem

(−16)(3/4)

Solution

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

(−16)(3/4)=(√(4,−16))3

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

√(4,−16)

3. Apply the definition of the imaginary unit i where i=√(,−1) We can factor the radicand.

√(4,−16)=√(4,16⋅(−1))

4. Simplify the root. The fourth root of 16 is 2 and the fourth root of −1 can be expressed using complex numbers.

√(4,16)⋅√(4,−1)=2⋅√(4,−1)

5. Use Euler's formula or De Moivre's Theorem to find the principal fourth root of −1 Since −1=e(i*π) the principal root is e(i*π/4)

e(i*π/4)=cos(π/4)+i*sin(π/4)

cos(π/4)+i*sin(π/4)=√(,2)/2+(i√(,2))/2

6. Multiply by 2 to find the value of the fourth root.

2⋅(√(,2)/2+(i√(,2))/2)=√(,2)+i√(,2)

7. Cube the result to find the final value.

(√(,2)+i√(,2))3=(√(,2)*(1+i))3

(√(,2))3⋅(1+i)3=2√(,2)⋅(1+3*i+3*i2+i3)

8. Simplify the powers of i where i2=−1 and i3=−i

2√(,2)⋅(1+3*i−3−i)=2√(,2)⋅(−2+2*i)

2√(,2)⋅(−2+2*i)=−4√(,2)+4*i√(,2)

Final Answer

(−16)(3/4)=−4√(,2)+4*i√(,2)

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