Worked examples
The question
Question: simplify √(−16).
√−16
No real number times itself is negative.
So we use a new number: i, the square root of minus 1.
So i squared is minus 1.
i = √(−1)
i² = −1
1 Split off the minus
−16 is 16 times minus 1.
= √16 × (−1)
We split the root: the root of 16, times the root of minus 1.
We split only to take out the minus 1.
= √16 × √−1
2 Use i
The square root of 16 is the positive number that times itself makes 16.
4 times 4 is 16.
And the square root of minus 1 is i.
= 4 × i
So √(−16) is 4i.
= 4i
3 Check
Let's check: square the answer.
4 squared is 16, and i squared is minus 1.
We get −16.
Check: (4i)2 = 16 × i2 = 16 × (−1) = −16
Let's look back.
First: split off the minus.
Then: use i.
Last: check.
▶ Watch it solved
The question
Question: work out (2 + 3i)(1 − i).
(2 + 3i)(1 − i)
No real number times itself is negative.
So we use a new number: i, the square root of minus 1.
So i squared is minus 1.
i = √(−1)
i² = −1
1 Multiply each part
Each part of the first bracket times each part of the second.
2 times 1 is 2.
2 times −i is −2i.
3i times 1 is 3i.
3i times −i is −3i².
2 × 1, 2 × (−i), 3i × 1, 3i × (−i)
We write all the parts.
= 2 − 2i + 3i − 3i2
2 Change i² to −1
i squared is minus 1.
So minus 3i² becomes 3.
Minus times minus is plus.
−3i² = −3 × (−1) = 3
Now we have 2 − 2i + 3i + 3.
= 2 − 2i + 3i + 3
3 Group like parts
Real parts together, i parts together.
Real parts: 2 + 3 = 5.
i parts: −2i + 3i = i.
2 + 3 = 5
−2i + 3i = i
So the answer is 5 + i.
= 5 + i
4 Check
Let's check: multiply the other way round, (1 − i) × (2 + 3i).
Each part times each part again, and i² is −1.
We get 5 + i again.
So the answer is right!
Check: (1 − i) × (2 + 3i) = 2 + 3i − 2i − 3i²
= 2 + 3i − 2i + 3 = 5 + i
Let's look back.
First: multiply each part.
Then: change i² to −1.
Then: group like parts.
Last: check.
▶ Watch it solved
The question
Question: work out (2 + i)(2 − i).
(2 + i)(2 − i)
No real number times itself is negative.
So we use a new number: i, the square root of minus 1.
So i squared is minus 1.
i = √(−1)
i² = −1
1 Multiply each part
Each part of the first bracket times each part of the second.
2 times 2 is 4.
2 times −i is −2i.
i times 2 is 2i.
i times −i is −i².
2 × 2, 2 × (−i), i × 2, i × (−i)
We write all the parts.
= 4 − 2i + 2i − i2
2 Change i² to −1
i squared is minus 1.
So minus i² becomes 1.
Minus times minus is plus.
−i² = −(−1) = 1
Now we have 4 − 2i + 2i + 1.
= 4 − 2i + 2i + 1
3 Group like parts
Real parts together, i parts together.
Real parts: 4 + 1 = 5.
i parts: −2i + 2i = 0.
4 + 1 = 5
−2i + 2i = 0
So the answer is 5.
The i parts cancel!
= 5
4 Check
Let's check: multiply the other way round, (2 − i) × (2 + i).
Each part times each part again, and i² is −1.
We get 5 again.
So the answer is right!
Check: (2 − i) × (2 + i) = 4 + 2i − 2i − i²
= 4 + 2i − 2i + 1 = 5
Let's look back.
First: multiply each part.
Then: change i² to −1.
Then: group like parts.
Last: check.
▶ Watch it solved
