You can take square roots from numbers as well as from terms. Square roots are also very important when solving equations. Taking the square root is defined as the inverse operation to squaring.
The Square Root
For some given positive number a, the square root b=a is that number with b2=(a)2=a. So when you square the square root b, then it becomes again a.
The square root b=a is always positive or 0.
The square root of a is sometimes also denoted as 2a. This is, because also higher-order roots3a, 4a, ... exist (these are NOT square roots). The numbers 3,4,… are also called indices, here.
The number a is sometimes also called radicand. It is always positive or 0.
Examples
4=2, because 22=4. Careful:(−2)×(−2)=4, but −2 is not the solution, because the square root of a number is always positive.
9=3, because 32=9.
81=9, because 92=81.
−3 does not exist, because the radicand is negative.
Square roots of terms
You can not only take roots of numbers, but also of terms. Also here, the radicand (= the term under the root) must not become negative. And just as with square roots of numbers, the square root of a term is always positive or 0.
Examples
5x+8
(a+2)2
−x−7
Domain of definition
When taking roots from terms, you have to make sure that the radicand is not negative. You can do this by carefully adjusting your domain of definition.
Square roots and absolute values
If there is a square term x2 under the square root x2, and you want to resolve the root, then you have to take the absolute value:
x2=∣x∣
This is because square roots must be positive and you want to obtain a positive result.
(x)2=x
How to handle roots of terms
Generally
Example 6x2
1. First determine the domain of definition for the radicand.
The radicand is 6x2. It never becomes negative because the variable x is squared.
Therefore, the domain of definition is all of R, i.e. all positive and negative numbers.
2. Take the root and then an absolute value.
6x2=6⋅x
Consider whether you can drop the absolute value.
It can be drop if the term inside the absolute value is positive or 0 for all numbers in the domain of definition.
If one were to use negative values for x (these are in the domain of definition), one would get the expression −6⋅x in the absolute value, which is not 6⋅x.
So you must not drop the absolute value.
Calculation rules
Calculation rule
Example
a⋅b=ab
ba=ba
a2=∣a∣
(a)2=a
a⋅a=a
a=a21
12=4⋅3=4⋅3=23
436=436=9=3
(−2)2=∣−2∣=2
(21)2=21
3⋅3=3
17=1721
Making denominators rational
If a number is given by ba, then you can expand this fraction with b to eliminate the root from the denominator. The calculation reads as follows:
ba=ba⋅bb=b⋅ba⋅b=bab
Taking square roots in equations
If you use roots to simplify equations, you have to be careful not to lose some solutions by accidentally "deleting information"! That's why you have to use the absolute value here too.
A simple example will illustrate this:
x2
=
4
↓
Take the root on both sides
x2
=
4
↓
Now take the root on both sides according to the above calculation rules (including absolute values).
∣x∣
=
2
↓
Here it is important not to forget the absloute value. If you resolve the absolute value, you get two sloutions.
x1
=
2
x2
=
−2
If you had not used the dashes, the solution would only have been x=2. So you would have lost the solution x=−2.