Further Maths Complex numbers: de Moivre Practice Questions

Read off the modulus and the argument

The base is already written in modulus-argument form.

Apply de Moivre's theorem

Raise the modulus to the power $3$ and multiply the argument by $3$.

State the answer in the form $a+bi$

This is the exact value of the power.

State the inductive assumption

Assume the result is true for the positive integer $n=k$.

Write the $\left(k+1\right)$th power as a product

Split off one factor so that the assumption can be used.

Substitute the inductive assumption

Replace the $k$th power by the assumed expression.

Select the correct final line

This is the statement for $n=k+1$, which completes the inductive step.

Apply de Moivre's theorem

The left-hand side is then expanded with the binomial theorem.

Take the real part of the binomial expansion

Equating parts isolates the multiple angle.

Remove the unwanted function with the Pythagorean identity

Only even powers occur, so the substitution is legitimate.

Collect like terms

This is the polynomial form of the multiple angle.

Check at $\theta=0$ to eliminate options

Any option that fails this test can be discarded immediately.

Quote de Moivre's theorem

This is the result quoted in the formula book; it holds for every integer $n$.

Select the matching option

This is the only option that is a correct identity.

Apply de Moivre's theorem

The left-hand side is then expanded with the binomial theorem.

Take the real part of the binomial expansion

Equating parts isolates the multiple angle.

Remove the unwanted function with the Pythagorean identity

Only even powers occur, so the substitution is legitimate.

Collect like terms

This is the polynomial form of the multiple angle.

Check at $\theta=0$ to eliminate options

Any option that fails this test can be discarded immediately.

Quote de Moivre's theorem

This is the result quoted in the formula book; it holds for every integer $n$.

Recall modulus-argument form

Here $r=\left|z\right|$ and $\theta=\arg z$.

Recall the power rule in modulus-argument form

The modulus is raised to the power and the argument is multiplied by it.

Recall Euler's relation

Exponential form is the compact way of writing modulus-argument form.

Recall the power rule in exponential form

A power of an exponential simply multiplies the exponent.

Select the matching option

This is the only option that is a correct identity.

Write the negative power as a reciprocal

A negative index means the reciprocal of the positive power.

Use de Moivre's theorem on the positive power

The theorem has already been proved for positive integers.

Multiply the numerator and the denominator by the conjugate

This is the standard way of dividing by a complex number.

Simplify the denominator

The denominator is the squared modulus, which equals $1$.

Note the modulus of the base

A unit modulus is what makes the denominator collapse to $1$.

Recall the conjugate of a unit complex number

The conjugate reflects the number in the real axis.

Rewrite using even and odd symmetry

This shows the answer is exactly the de Moivre form with $-n$ in place of $n$.

Conclude that the theorem holds for negative integers

The result therefore holds for every integer index.

Quote de Moivre's theorem

This is the result quoted in the formula book; it holds for every integer $n$.

Recall modulus-argument form

Here $r=\left|z\right|$ and $\theta=\arg z$.

Recall the power rule in modulus-argument form

The modulus is raised to the power and the argument is multiplied by it.

Recall Euler's relation

Exponential form is the compact way of writing modulus-argument form.

Recall the power rule in exponential form

A power of an exponential simply multiplies the exponent.

Recall the effect of a power on the modulus

Moduli multiply, so a power of $z$ raises the modulus to that power.

Select the correct simplified form

This is the reciprocal written in de Moivre form.