For z = 3 − 4i, compute z^5 in a+bi form.

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For z = 3 − 4i, compute z^5 in a+bi form.

Raising a complex number to a power can be done cleanly by multiplying step by step. Start with z = 3 − 4i and multiply successively to build up to the fifth power, keeping track of real and imaginary parts at each step. Compute the square first: z^2 = (3 − 4i)^2 = 9 − 24i + 16i^2 = 9 − 24i − 16 = −7 − 24i. Then the cube: z^3 = z^2 · z = (−7 − 24i)(3 − 4i) = −21 + 28i − 72i + 96i^2 = −21 − 44i − 96 = −117 − 44i. Next the fourth power: z^4 = z^3 · z = (−117 − 44i)(3 − 4i) = −351 + 468i − 132i + 176i^2 = −351 + 336i − 176 = −527 + 336i. Finally the fifth power: z^5 = z^4 · z = (−527 + 336i)(3 − 4i) = −1581 + 2108i + 1008i − 1344i^2 = −1581 + 3116i + 1344 = −237 + 3116i. So in a+bi form, z^5 = −237 + 3116i.

Raising a complex number to a power can be done cleanly by multiplying step by step. Start with z = 3 − 4i and multiply successively to build up to the fifth power, keeping track of real and imaginary parts at each step.

Compute the square first: z^2 = (3 − 4i)^2 = 9 − 24i + 16i^2 = 9 − 24i − 16 = −7 − 24i.

Then the cube: z^3 = z^2 · z = (−7 − 24i)(3 − 4i) = −21 + 28i − 72i + 96i^2 = −21 − 44i − 96 = −117 − 44i.

Next the fourth power: z^4 = z^3 · z = (−117 − 44i)(3 − 4i) = −351 + 468i − 132i + 176i^2 = −351 + 336i − 176 = −527 + 336i.

Finally the fifth power: z^5 = z^4 · z = (−527 + 336i)(3 − 4i) = −1581 + 2108i + 1008i − 1344i^2 = −1581 + 3116i + 1344 = −237 + 3116i.

So in a+bi form, z^5 = −237 + 3116i.

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