Use DeMoivre's Theorem to calculate the following expression. Write the exact answer in the form found using Euler's Formula, |z|eiθ|𝑧|𝑒𝑖𝜃. Do not round. Make sure that the argument of your answer lies in the interval [0,2π)[0,2𝜋). [2(cos(π7)+isin(π7))]5
Question
Use DeMoivre's Theorem to calculate the following expression. Write the exact answer in the form found using Euler's Formula, |z|eiθ|𝑧|𝑒𝑖𝜃. Do not round. Make sure that the argument of your answer lies in the interval [0,2π)[0,2𝜋). [2(cos(π7)+isin(π7))]5
Solution
To solve this problem, we will use DeMoivre's Theorem, which states that for any complex number in polar form, (r(cos θ + i sin θ))^n = r^n (cos(nθ) + i sin(nθ)).
Given the expression [2(cos(π/7) + i sin(π/7))]^5, we can identify r = 2 and θ = π/7.
Step 1: Calculate r^n r^n = 2^5 = 32
Step 2: Calculate nθ nθ = 5 * π/7 = 5π/7
Step 3: Substitute r^n and nθ into the formula 32(cos(5π/7) + i sin(5π/7))
So, the result of the expression [2(cos(π/7) + i sin(π/7))]^5 using DeMoivre's Theorem is 32(cos(5π/7) + i sin(5π/7)).
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