Question

Simplify the radical:=" \sqrt{5x^4} " align="absmiddle" class="latex-formula">/$\sqrt{10x}$

Answer 1

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:

$\sqrt{ab} = \sqrt{a} \sqrt{b}$

First, we will split up the top.

$\sqrt{5x^4} = \sqrt{5} \sqrt{x^4}$

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 $x^2$

$\sqrt{5} \sqrt{x^4} = \sqrt{5} \sqrt{x^2} \sqrt{x^2}$

Now we can simplify.

$\sqrt{5} \sqrt{x^2} \sqrt{x^2} = \sqrt{5} x * x$

$\sqrt{5} x * x = \sqrt{5}x^2$

The top is done, now for the bottom one.

$\sqrt{10x} = \sqrt{10} \sqrt{x}$

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.

$\frac{\sqrt{5}x^2}{\sqrt{10} \sqrt{x}}$

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:

$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

Using this, simplify the expression further.

$\frac{\sqrt{5}x^2}{\sqrt{10} \sqrt{x}} = \sqrt{\frac{5}{10}} \frac{x^2}{\sqrt{x}}$

$\sqrt{\frac{5}{10}} \frac{x^2}{\sqrt{x}} = \sqrt{\frac{1}{2}} \frac{x^2}{\sqrt{x}}$

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