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Simplifying square roots
https://www.khanacademy.org/math/algebra/exponent-equations/exponent-propertiesalgebra/v/simplifying-square-roots
Simplifying Square Roots
To simplify a square root: make the number inside the square root as small as possible (but still
a whole number):
Example: √8 is simpler as 2√2
Get your calculator and check if you want: they are both the same value!
This is the useful rule to remember:
And this is how to use it:
Example (continued)
√8 = √(4×2) = √4 × √2 = 2√2
(Because the square root of 4 is 2)
Here is another example:
Example: simplify √12
12 is 4 times 3:
√12 = √(4 × 3)
Use the rule:
√(4 × 3) = √4 × √3
And the square root of 4 is 2:
√4 × √3 = 2√3
So √12 is simpler as 2√3
And here is how to simplify in one line:
Example: simplify √18
√18 = √(9 × 2) = √9 × √2 = 3√2
It often helps to factor the numbers (into prime numbers is best):
Example: simplify √6 × √15
First we can combine the two numbers:
√6 × √15 = √(6 × 15)
Then we 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) × √(2 × 5) = 3√10
Fractions
There is a similar rule for fractions:
Example: simplify √30 / √10
First we can combine the two numbers:
√30 / √10 = √(30 / 10)
Then simplify:
√(30 / 10) = √3
A Harder Example
Example: simplify (√20 × √5) / √2
See if you can follow the steps:
(√20 × √5)/√2
(√(2 × 2 × 5) × √5)/√2
(√2 × √2 × √5 × √5)/√2
√2 × √5 × √5
√2 × 5
5√2
Surds
If you can't simplify a number to remove a square root (or cube root etc) then it is a
surd.
Example: √2 (square root of 2) can't be simplified further so it is a surd
Example: √4 (square root of 4) can be simplified (to 2), so it is not a surd!
Have a look at some more examples:
Number Simplifed
As a Decimal
Surd or
not?
√2
√2
1.4142135...(etc)
Surd
√3
√3
1.7320508...(etc)
Surd
√4
2
2
Not a surd
√(1/4)
1/2
0.5
Not a surd
3√(11)
3√(11)
2.2239800...(etc)
Surd
3√(27)
3
3
Not a surd
5√(3)
5√(3)
1.2457309...(etc)
Surd
As you can see, the surds have a decimal which goes on forever without repeating,
and are actually Irrational Numbers.
In fact "Surd" used to be another name for "Irrational", but
it is now used for aroot that is irrational.
How did we get the word "Surd" ?
Well around 820 AD al-Khwarizmi (the Persian guy who
we get the name "Algorithm" from) called irrational
numbers "'inaudible" ... this was later translated to the
Latin surdus ("deaf" or "mute")
Conclusion

If it is a root and irrational, it is a surd.

But not all roots are surds.
How to rationalize a denominator
https://www.khanacademy.org/math/algebra/exponent-equations/exponent-propertiesalgebra/v/how-to-rationalize-a-denominator
Rationalize the Denominator
"Rationalising the denominator" is when you move a
root (like a square root or cube root) from the bottom of a
fraction to the top.
1.1
Oh No! An Irrational Denominator!
The bottom of a fraction is called the denominator.
Many roots, such as √2 and √3, are irrational, but numbers like 2 and
3 are rational.
Example:
has an Irrational Denominator
To be in "simplest form" the denominator should not be irrational!
Fixing it (by making the denominator rational)
is called "Rationalizing the Denominator"
Note: there is nothing wrong with an irrational denominator, it still works, but it is
not "simplest form" and so can cost you marks. And removing them may help
you solve an equation, so you should learn how.
So ... how do you do it?
Multiply Both Top and Bottom by a Root
Sometimes you can just multiply both top and bottom by a root:
Example:
has an Irrational Denominator. Let's fix it.
Multiply top and bottom by the square root of 2, because: √2 × √2 = 2:
Now the denominator has a rational number (=2). Done!
Note: It is ok to have an irrational number in the top (numerator) of a fraction.
Multiply Both Top and Bottom by the Conjugate
There is another special way to move a square root from the bottom of a fraction to
the top ... you multiply both top and bottom by the conjugate of the
denominator.
The conjugate is where you change the sign in the middle of two terms:
Example Expression
Its Conjugate
x2 - 3
x2 + 3
Another Example
Its Conjugate
a + b3
a - b3
It works because when you multiply something by its conjugate you
get squares like this:
Here is how you do it:
Example: here is a fraction with an "irrational denominator":
How can we move the square root of 2 to the top?
Answer: Multiply both top and bottom by the conjugate of 3-√2 (this
will not change the value of the fraction), like this:
(Did you see how the denominator became a2-b2 ?)
Useful
So try to remember these little tricks, it may help you solve an equation one day!