Simplify the radical:[tex] \sqrt{5x^4} [/tex]/[tex] \sqrt{10x} [/tex]

Split up the top and bottom into factors and cancel out any common factors.

While we do these problems, it is important to remember this rule:

[tex]\sqrt{ab} = \sqrt{a} \sqrt{b}[/tex]

First, we will split up the top.

[tex]\sqrt{5x^4} = \sqrt{5} \sqrt{x^4}[/tex]

We can't do anything with the square root of 5 because it is in it's simplest form. But, we can still split up the square root of [tex]x^2[/tex]

[tex]\sqrt{5} \sqrt{x^4} = \sqrt{5} \sqrt{x^2} \sqrt{x^2}[/tex]

Now we can simplify.

[tex]\sqrt{5} \sqrt{x^2} \sqrt{x^2} = \sqrt{5} x * x[/tex]

[tex]\sqrt{5} x * x = \sqrt{5}x^2[/tex]

The top is done, now for the bottom one.

[tex]\sqrt{10x} = \sqrt{10} \sqrt{x}[/tex]

We can't simplify either the square root of 10 or the square root of x.
So, the top is done, too.

Find any common factors and cancel them out.

[tex]\frac{\sqrt{5}x^2}{\sqrt{10} \sqrt{x}}[/tex]

Well, I can't find any common factors to cancel out, but, we can still simplify it further.

There is another law regarding radical expressions:

[tex]\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}[/tex]

Using this, simplify the expression further.

[tex]\frac{\sqrt{5}x^2}{\sqrt{10} \sqrt{x}} = \sqrt{\frac{5}{10}} \frac{x^2}{\sqrt{x}}[/tex]

[tex]\sqrt{\frac{5}{10}} \frac{x^2}{\sqrt{x}} = \sqrt{\frac{1}{2}} \frac{x^2}{\sqrt{x}}[/tex]

So, now it is in it's simplest form.