Simplifying Square Roots

To simplify a square root: make the number inside the square root as small-scale equally possible (only nevertheless a whole number):

Case: √12 is simpler equally 2√3

Go your reckoner and bank check if you lot want: they are both the same value!

Hither is the rule: when a and b are not negative

√(ab) = √a × √b

And here is how to use information technology:

Instance: simplify √12

12 is 4 times 3:

√12 = √(4 × 3)

Utilize the rule:

√(4 × 3) = √4 × √3

And the square root of iv is 2:

√4 × √three = 2√3

Then √12 is simpler as 2√3

Another example:

Example: simplify √eight

√8 = √(four×2) = √iv × √2 = ii√2

(Because the square root of four is two)

And another:

Example: simplify √18

√xviii = √(9 × two) = √9 × √2 = 3√2

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

Case: simplify √half dozen × √15

First we tin can combine the two numbers:

√half dozen × √fifteen = √(6 × fifteen)

Then nosotros cistron them:

√(half dozen × fifteen) = √(2 × three × three × five)

So we see ii 3s, and decide to "pull them out":

√(2 × 3 × 3 × 5) = √(3 × 3) × √(2 × five) = 3√10

Fractions

There is a similar rule for fractions:

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

Example: simplify √thirty / √10

First nosotros can combine the 2 numbers:

√30 / √10 = √(30 / 10)

So simplify:

√(30 / x) = √3

Some Harder Examples

Example: simplify √xx × √5 √2

See if you lot can follow the steps:

√20 × √5 √2

√(ii × two × 5) × √5 √2

√2 × √ii × √v × √5 √2

√ii × √v × √5

√2 × 5

5√2

Instance: simplify 2√12 + 9√three

Start simplify 2√12:

2√12 = 2 × 2√three = 4√3

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

4√3 + 9√3 = (4+9)√3 = 13√3

Surds

Notation: a root we can't simplify farther is chosen a Surd. So √3 is a surd. Merely √4 = 2 is not a surd.