Simplifying Square Roots

To simplify a foursquare root: brand the number inside the foursquare root as small-scale as possible (but nevertheless a whole number):

Example: √12 is simpler as 2√3

Go your estimator and check if you desire: they are both the same value!

Here is the rule: when a and b are non negative

√(ab) = √a × √b

And here is how to use it:

Case: simplify √12

12 is 4 times 3:

√12 = √(4 × 3)

Apply the dominion:

√(4 × three) = √4 × √3

And the square root of 4 is ii:

√four × √3 = 2√3

So √12 is simpler as ii√3

Some other case:

Example: simplify √viii

√eight = √(4×2) = √iv × √2 = 2√2

(Considering the square root of 4 is 2)

And another:

Example: simplify √xviii

√18 = √(ix × 2) = √9 × √2 = 3√ii

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

Example: simplify √half-dozen × √15

First we can combine the 2 numbers:

√6 × √xv = √(6 × 15)

So nosotros cistron them:

√(half-dozen × 15) = √(2 × 3 × three × v)

Then nosotros see two 3s, and decide to "pull them out":

√(2 × 3 × 3 × v) = √(3 × iii) × √(2 × v) = iii√10

Fractions

There is a similar dominion for fractions:

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

Example: simplify √30 / √10

Showtime nosotros can combine the two numbers:

√thirty / √10 = √(xxx / 10)

Then simplify:

√(30 / 10) = √three

Some Harder Examples

Example: simplify √20 × √5 √two

Come across if you can follow the steps:

√20 × √five √2

√(2 × 2 × five) × √5 √two

√two × √2 × √v × √5 √2

√2 × √5 × √five

√2 × 5

v√2

Example: simplify 2√12 + nine√3

First simplify two√12:

2√12 = 2 × 2√three = 4√3

Now both terms have √3, we can add them:

4√3 + ix√three = (4+9)√three = 13√3

Surds

Note: a root nosotros can't simplify further is called a Surd. And so √3 is a surd. But √iv = 2 is not a surd.