Step by step
The question
Question: work out √12 × √15.
√12 × √15
1 Multiply
Rule: two roots times each other make one root of the two numbers times each other.
√a × √b = √(a × b)
So it is the root of 12 times 15.
= √12 × 15
Careful!
This works for times, not for plus.
√9 + √16 is 3 + 4 = 7, but √25 is 5.
Careful: not √12 + √15 = √27 ✗
only for ×: √9 + √16 = 7, √25 = 5
12 times 15 is 180.
= √180
2 Simplify the root
Now we make √180 simpler, like in the lesson on square roots.
81 does not go into 180.
64 does not go into 180.
49 does not go into 180.
The biggest square that goes into 180 is 36: 180 divided by 36 is 5.
Square numbers: 4, 9, 16, 25, 36, 49, 64, 81
180 ÷ 36 = 5 ✓
So √180 is √36 times √5.
= √36 × √5
√36 is 6, because 6 times 6 is 36.
6 × 6 = 36
So we get 6√5.
So √12 × √15 = 6√5.
= 6√5
3 Check
Let's check with a calculator.
The start is about 13.42.
Start: √12 × √15 ≈ 13.42
The answer is about 13.42 too.
So it is right!
Answer: 6√5 ≈ 13.42
Let's look back.
First: multiply.
Then: simplify the root.
Last: check.
Answer
= 6√5