Worked examples
The question
Question: work out √2 × √8.
√2 × √8
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 2 times 8.
= √2 × 8
Careful!
This works for times, not for plus.
√9 + √16 is 3 + 4 = 7, but √25 is 5.
Careful: not √2 + √8 = √10 ✗
only for ×: √9 + √16 = 7, √25 = 5
2 times 8 is 16.
= √16
2 Simplify the root
16 is a square number: 4 times 4 is 16.
So √16 is 4.
4 × 4 = 16
So we get 4.
So √2 × √8 = 4.
= 4
3 Check
Let's check with a calculator.
The start is about 4.00.
Start: √2 × √8 ≈ 4.00
The answer is about 4.00 too.
So it is right!
Answer: 4 ≈ 4.00
Let's look back.
First: multiply.
Then: simplify the root.
Last: check.
▶ Watch it solved
The question
Question: work out 2√3 × 5√6.
2√3 × 5√6
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.
= 2 × 5 × √3 × √6
2 times 5 is 10.
2 × 5 = 10
So it is 10 times the root of 3 times 6.
= 10√3 × 6
Careful!
This works for times, not for plus.
√9 + √16 is 3 + 4 = 7, but √25 is 5.
Careful: not √3 + √6 = √9 ✗
only for ×: √9 + √16 = 7, √25 = 5
3 times 6 is 18.
= 10√18
2 Simplify the root
Now we make √18 simpler, like in the lesson on square roots.
The biggest square that goes into 18 is 9: 18 divided by 9 is 2.
Square numbers: 4, 9
18 ÷ 9 = 2 ✓
So √18 is √9 times √2.
= 10 × √9 × √2
√9 is 3, because 3 times 3 is 9.
3 × 3 = 9
10 times 3 is 30.
10 × 3 = 30
So we get 30√2.
So 2√3 × 5√6 = 30√2.
= 30√2
3 Check
Let's check with a calculator.
The start is about 42.43.
Start: 2√3 × 5√6 ≈ 42.43
The answer is about 42.43 too.
So it is right!
Answer: 30√2 ≈ 42.43
Let's look back.
First: multiply.
Then: simplify the root.
Last: check.
▶ Watch it solved
The question
Question: work out √20 × √15.
√20 × √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 20 times 15.
= √20 × 15
Careful!
This works for times, not for plus.
√9 + √16 is 3 + 4 = 7, but √25 is 5.
Careful: not √20 + √15 = √35 ✗
only for ×: √9 + √16 = 7, √25 = 5
20 times 15 is 300.
= √300
2 Simplify the root
Now we make √300 simpler, like in the lesson on square roots.
144 does not go into 300.
121 does not go into 300.
The biggest square that goes into 300 is 100: 300 divided by 100 is 3.
Square numbers: 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
300 ÷ 100 = 3 ✓
So √300 is √100 times √3.
= √100 × √3
√100 is 10, because 10 times 10 is 100.
10 × 10 = 100
So we get 10√3.
So √20 × √15 = 10√3.
= 10√3
3 Check
Let's check with a calculator.
The start is about 17.32.
Start: √20 × √15 ≈ 17.32
The answer is about 17.32 too.
So it is right!
Answer: 10√3 ≈ 17.32
Let's look back.
First: multiply.
Then: simplify the root.
Last: check.
▶ Watch it solved
