Simplifying Square Roots

To simplify a square root: make the number inside the foursquare root every bit small as possible (but notwithstanding a whole number):

Example: √12 is simpler equally 2√3

Get your calculator and cheque if you want: they are both the same value!

Here is the dominion: when a and b are not negative

√(ab) = √a × √b

And here is how to use it:

Instance: simplify √12

12 is 4 times 3:

√12 = √(4 × three)

Utilise the rule:

√(four × three) = √iv × √iii

And the square root of four is 2:

√4 × √iii = 2√3

And so √12 is simpler as ii√3

Some other example:

Instance: simplify √8

√eight = √(4×2) = √4 × √ii = 2√two

(Because the square root of four is 2)

And some other:

Case: simplify √18

√18 = √(nine × 2) = √9 × √2 = 3√2

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

Example: simplify √6 × √fifteen

Get-go nosotros can combine the two numbers:

√6 × √fifteen = √(6 × 15)

And then nosotros factor them:

√(6 × 15) = √(2 × 3 × 3 × 5)

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

√(2 × 3 × 3 × 5) = √(3 × 3) × √(ii × 5) = three√x

Fractions

There is a similar rule for fractions:

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

Instance: simplify √thirty / √x

First nosotros can combine the 2 numbers:

√30 / √ten = √(30 / 10)

Then simplify:

√(30 / 10) = √3

Some Harder Examples

Instance: simplify √twenty × √v √two

See if you can follow the steps:

√twenty × √v √ii

√(2 × 2 × 5) × √v √2

√2 × √2 × √5 × √5 √two

√2 × √5 × √5

√2 × 5

5√2

Example: simplify two√12 + ix√iii

First simplify 2√12:

ii√12 = 2 × 2√3 = 4√3

At present both terms have √iii, we can add together them:

4√3 + ix√three = (4+9)√3 = xiii√3

Surds

Note: a root nosotros can't simplify farther is called a Surd. So √three is a surd. But √4 = 2 is not a surd.