Step by step
The question
Question: work out 4√6 × 2√2.
4√6 × 2√2
1 Multiply
Rule: two roots times each other make one root of the two numbers times each other.
√a × √b = √(a × b)
Times can go in any order.
So we put the numbers outside together, and the roots together.
= 4 × 2 × √6 × √2
4 times 2 is 8.
4 × 2 = 8
So it is 8 times the root of 6 times 2.
= 8√6 × 2
Careful!
This works for times, not for plus.
√9 + √16 is 3 + 4 = 7, but √25 is 5.
Careful: not √6 + √2 = √8 ✗
only for ×: √9 + √16 = 7, √25 = 5
6 times 2 is 12.
= 8√12
2 Simplify the root
Now we make √12 simpler, like in the lesson on square roots.
The biggest square that goes into 12 is 4: 12 divided by 4 is 3.
Square numbers: 4
12 ÷ 4 = 3 ✓
So √12 is √4 times √3.
= 8 × √4 × √3
√4 is 2, because 2 times 2 is 4.
2 × 2 = 4
8 times 2 is 16.
8 × 2 = 16
So we get 16√3.
So 4√6 × 2√2 = 16√3.
= 16√3
3 Check
Let's check with a calculator.
The start is about 27.71.
Start: 4√6 × 2√2 ≈ 27.71
The answer is about 27.71 too.
So it is right!
Answer: 16√3 ≈ 27.71
Let's look back.
First: multiply.
Then: simplify the root.
Last: check.
Answer
= 16√3