Example: find w so that

**Method 1: calculator**

Convert to r ∠ θ form: w^{2} ≈ 34 cis 2.06

Mentally, square-root the modulus and halve the argument

w ≈ √34 cis 1.03

Convert back to a+bi form: w ≈ 3+5i (round your answer back to a neat value, to correct the calculator error introduced when halving a decimal approximation of the argument)

Check that you’ve rounded back to an exactly correct answer:

(3+5i)^{2} = -16 + 30i

Write your answer: w = ± (3+5i)

(See below for a tweak to this method)

**Method 2: Equate modulus and real part**

Let square root = w = a + bi

Then

Solve those simultaneous equations for a^{2} and b^{2}

a^{2}=9 and b^{2}=25

Only problem now is to work out the valid signs for a and b. . In this case a and b must have the same sign (both negative or both positive) because . (If then a and b would have to have opposite signs, one positive, one negative).

So w = ± (3+5i)

**Method 3: Equate real and imaginary parts** (the method in the old Edexcel textbook)

Let square root = w = a + bi

Then

Substitute from the second equation into the first

Multiply up to get a quadratic equation in

Solve: a^{2} = 9, so a = ± 3, and b = ± 5

So w = ± (3+5i)

**Method 1 (calculator) tweaked**

You can minimise the approximation problem in the calculator method by using the calculator rather than your head to halve the argument. As in these images (poor quality, sorry):

This will usually give you the exact answer in the exact form you want it. The likely exception is if you’re finding something like √(−1 + 2√2 i), where the answer is (1 + √2 i).

The calculator will give you (1 + 1.414213562 i), and it won’t convert that to (1 + √2 i) even if you press the S-D key.

Workaround: if you get an answer for real or imaginary part of the square root which isn’t a whole number, square your answer to see if it’s an exact square root of a whole number (in A level, it pretty much always will be).

It’s worth memorising the facts that

√2 ≈ 1.414

√3 ≈ 1.732

√5 ≈ 2.236

Often come in useful, like it comes in useful to be able to remember your way home from school, or to a friend’s house, without having to look at Google Maps.