Simplifying Square Roots

To simplify a square root: make the number inside the square root every bit pocket-size as possible (but still a whole number):

Example: √12 is simpler as ii√3

Get your calculator and check if you desire: they are both the same value!

Hither is the dominion: when a and b are non negative

√(ab) = √a × √b

And here is how to use information technology:

Example: simplify √12

12 is 4 times 3:

√12 = √(4 × iii)

Utilize the rule:

√(4 × 3) = √four × √3

And the square root of iv is 2:

√4 × √three = two√3

So √12 is simpler equally 2√3

Another example:

Example: simplify √eight

√viii = √(4×two) = √4 × √2 = 2√2

(Because the square root of 4 is 2)

And another:

Example: simplify √18

√18 = √(9 × two) = √nine × √2 = three√2

It often helps to factor the numbers (into prime numbers is best):

Example: simplify √six × √15

Start we can combine the two numbers:

√vi × √15 = √(half dozen × 15)

Then we factor them:

√(vi × 15) = √(2 × iii × 3 × 5)

And so we see 2 3s, and make up one's mind to "pull them out":

√(two × 3 × 3 × five) = √(three × 3) × √(2 × v) = 3√10

Fractions

There is a similar rule for fractions:

root a / root b = root (a / b)

Example: simplify √30 / √10

First we can combine the 2 numbers:

√30 / √10 = √(30 / 10)

Then simplify:

√(30 / ten) = √3

Some Harder Examples

Example: simplify √20 × √five √2

See if y'all can follow the steps:

√20 × √5 √two

√(2 × two × 5) × √five √two

√2 × √ii × √five × √five √2

√2 × √5 × √5

√two × 5

5√two

Case: simplify 2√12 + 9√three

First simplify 2√12:

2√12 = two × 2√3 = 4√3

Now both terms take √three, nosotros tin can add them:

4√3 + ix√3 = (four+nine)√3 = xiii√3

Surds

Note: a root we tin't simplify further is called a Surd. So √3 is a surd. But √iv = 2 is not a surd.