2023TrialShort answer

Q11 (a)

2 marks

Question

By first writing $\sqrt{3}+i$ and $\sqrt{3}-i$ in exponential form, express

(\sqrt{3}+i)^{12}+(\sqrt{3}-i)^{12}

in the form of $a^b$ , where $a$ and $b$ are integers.

Try the question before revealing the answer.

Solution

\sqrt{3}+i=2e^{i\pi/6}

\sqrt{3}-i=2e^{-i\pi/6}

(\sqrt{3}+i)^{12}+(\sqrt{3}-i)^{12}=(2e^{i\pi/6})^{12}+(2e^{-i\pi/6})^{12}

=2^{12}e^{i2\pi}+2^{12}e^{-i2\pi}

=2^{12}+2^{12}

=2^{13}

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Complex Numbers

Other representations of complex numbersUse Euler's formula and convert between Cartesian, polar and exponential forms to solve problems.

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