Simplifying Square Roots

To simplify a square root: make the number within the square root as small every bit possible (just still a whole number):

Example: √12 is simpler equally 2√3

Go your calculator and bank check if yous want: they are both the same value!

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

√(ab) = √a × √b

And here is how to use it:

Example: simplify √12

12 is iv times 3:

√12 = √(4 × 3)

Use the dominion:

√(4 × 3) = √4 × √three

And the square root of four is 2:

√4 × √3 = 2√3

So √12 is simpler as ii√3

Another case:

Example: simplify √8

√viii = √(four×two) = √four × √2 = 2√2

(Because the square root of 4 is 2)

And another:

Example: simplify √18

√18 = √(9 × 2) = √9 × √two = 3√2

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

Example: simplify √6 × √15

Outset we tin combine the two numbers:

√6 × √fifteen = √(half-dozen × 15)

And then we factor them:

√(vi × 15) = √(2 × 3 × 3 × 5)

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

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

Fractions

There is a similar dominion for fractions:

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

Instance: simplify √30 / √10

Offset we can combine the 2 numbers:

√30 / √x = √(30 / 10)

Then simplify:

√(30 / 10) = √3

Some Harder Examples

Case: simplify √xx × √5 √2

See if you tin follow the steps:

√twenty × √v √ii

√(2 × ii × v) × √5 √2

√two × √2 × √5 × √5 √two

√ii × √v × √v

√2 × 5

5√2

Example: simplify 2√12 + 9√3

First simplify 2√12:

2√12 = 2 × 2√three = 4√iii

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

4√iii + ix√3 = (4+9)√3 = 13√iii

Surds

Note: a root nosotros can't simplify further is called a Surd. So √3 is a surd. But √4 = two is not a surd.