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Simplify Square Root
We can "simplify" a square root like 12 to 2 3. To do this, we need to learn the Product Rule of Square Root first.
Product Rule of Square Root
Let's review an important rule we learned earlier. To reduce a fraction, we can do:
= /^ โ
but we cannot do:
= /^ +
=^ +
For fraction reduction, multiplication/division make our life easy, while addition/subtraction make our life hard.
There is a similar situation for square root. We can do:
9 โ
4 = 9 โ
4 and 9
but we cannot do:
16 + 9 = 16 + 9 or 16 โ 9 = 16 โ 9
Don't memorize these as rules. Instead, try these operations on scratch paper and use a calculator to
verify the results. For example, for 16 + 9 = 16 + 9 , the left side gives 5, but the right side gives 7. This is how you know you may not do this.
On the other hand, for 9 โ
4 = 9 โ
4 , both sides give 6. We call this the Product Rule of Square Root :
a โ
b = a โ
b
We will use this rule to simplify square roots.
Simplifying Square Root
With a calculator, we can see:
This implies 12 and 2 3 are equivalent. I will show you why. Let's observe a pattern first:
x โ
x = x = x
2
There are two methods to simplify square root. Here is Method 1 :
12 = 2 โ
2 โ
3 = 2 3
- First, we factor 12 into 2 โ
2 โ
3. You might need to go to MTH20 content and review how to build a factor tree for a number.
- Next, since we have a pair of 2's inside the square root, we can take them out, and they become one 2 outside the square root symbol.
- Since the 3 is alone inside the square root symbol, it has to stay inside.
Here is Method 2 :
12 = 4 โ
3 = 4 โ
3 = 2 3
- First, we see which square number (bigger than 1) goes into 12. Think of each one in sequence: 4, 9, 16, ... In this case, 4 goes into 12.
- Next, break 12 into 4 โ
3.
- Use the Product Rule of Square Root to break 4 โ
3 into 4 โ
3.
- Finally, change 4 into 2.
You can choose which method to use. Method 1 always works, but it takes longer to build a prime factor tree. Method 2 is faster, but sometimes it's hard to see a certain square number goes into a given number.
Next, prime factor the number inside and simplify:
6 โ
30 = 180 = 2 โ
2 โ
3 โ
3 โ
5 = 2 โ
3 5 = 6 5
Unless you can see the square number 36 goes into 180:
6 โ
30 = 180 = 36 โ
5 = 36 โ
5 = 6 5
[ Example 4 ] Simplify 2 6 โ
5 30
[ Solution ] We separate numbers inside and outside the square roots:
2 6 โ
5 30 = 2 โ
5 โ
6 โ
30 = 10 180
Then, by what we did in Example 3, we can further simplify this expression:
10 180 = 10 โ
6 5 = 60 5
You should grow "suspicious" when you see numbers like 12, 24, 50. These numbers have square numbers as a factor. For example:
- 4 goes into these numbers: 8, 12, 20, 24, 28, 32, 40, ...
- 9 goes into these numbers: 18, 27, 45, 54, 63, ...
- 16 goes into these numbers: 32, 48, 80, 96, ...
- 25 goes into these numbers: 50, 75, 125, 150, ...
- ...
When these numbers are inside a square root, we can (and must) simplify them.
If a certain property works for multiplication, most of the time it also applies for division. We will learn another property of radicals next.
Quotient Rule of Square Root
It's important to connect knowledge. Earlier, we learned negative exponents:
x
x โ^1 =^1
A fraction line means division. This implies:
= ab โ^1 b
a
In other words, a fraction means division, or multiplication of the denominator to the negative first power. In most cases, we can change a multiplication into division, or vice versa. This is why when a property applies for multiplication, usually it applies to division.
We learned the Product Rule of Square Root:
a โ
b = a โ
b
Here is Quotient Rule of Square Root :
b
a b
a (^) =
[ Example 5 ] Simplify the following radicals.
Note that in the second problem, we first used the product rule, and then the quotient rule.
[ Example 6 ] Simplify the radical:
Be very careful: This radical can be further simplified:
