In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form.(1 + i)⁵

DeMoivre's Theorem states that for any complex number expressed in polar form as r(cos θ + i sin θ), the nth power of the complex number can be calculated as r^n (cos(nθ) + i sin(nθ)). This theorem simplifies the process of raising complex numbers to powers and is essential for solving problems involving complex exponentiation.

The polar form of a complex number is a way of expressing it using its magnitude (r) and angle (θ) instead of its rectangular coordinates (a + bi). The magnitude is calculated as r = √(a² + b²), and the angle is found using θ = arctan(b/a). This representation is crucial for applying DeMoivre's Theorem effectively.

The rectangular form of a complex number is expressed as a + bi, where a is the real part and b is the imaginary part. Converting the result from polar form back to rectangular form involves using the cosine and sine values to find the real and imaginary components, which is necessary for presenting the final answer in the required format.