This problem is an application of the Geometric mean theorem. It says that
[tex]\frac{6}{x}=\frac{x}{4}[/tex]
Comment: In other words, it says that the length of BC (x) is the geometric mean between the lengths of AC and DC.
Then,
[tex]x^2=6\cdot4=24[/tex][tex]x=\sqrt[]{24}=2\cdot\sqrt[]{6}[/tex]
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Let's talk a little about the simplest radical form of a square root
[tex]\sqrt[]{a}[/tex]
The first step to finding it is to write the number within the root as a product of prime powers, such product is called its integer factorization. Let's do that for 24:
Then, the integer factorization of 24 is
[tex]24=2^3\cdot3[/tex]
Thus,
[tex]\sqrt[]{24}=\sqrt[]{2^3\cdot3}[/tex]
The idea now is to take out of the root all we can. The rule is that we can only take out powers of 2 (for our root is a square root). In the expression
[tex]2^3\cdot3[/tex]
There is only one power of 2, within 2^3. We can write it as
[tex]2^2\cdot2\cdot3[/tex]
How are we going to take out it? We are going to take out the base of the power, which is 2 in this case. Then,
[tex]\sqrt[]{24}=2\cdot\sqrt[]{2\cdot3}=2\cdot\sqrt[]{6}[/tex]
In simple terms, the simplest radical form of a root is what results after taking out the root all that can be taken out.
