Respuesta :

Answer:

[tex]2+3\sqrt{5}[/tex]

Step-by-step explanation:

Hi!

This is a lot of radicals, so it may seem tricky. Let's begin by simplifying all of the radicals that can be simplified.

We can see that [tex]\sqrt{75}[/tex] can be simplified.

Taking prime factorization:

[tex]=\sqrt{5^2\cdot \:3}[/tex]

which is:

[tex]=5\sqrt{3}[/tex]

So an updated equation is [tex]\left(\sqrt{5}+\sqrt{3}-\sqrt{15}\right)\left(\sqrt{5}-\sqrt{3}\right)+5\sqrt{3}[/tex].

Now, we need to expand this:

First, let's expand the parenthesis:

[tex]\sqrt{5}\sqrt{5}+\sqrt{5}\left(-\sqrt{3}\right)+\sqrt{3}\sqrt{5}+\sqrt{3}\left(-\sqrt{3}\right)+\left(-\sqrt{15}\right)\sqrt{5}+\left(-\sqrt{15}\right)\left(-\sqrt{3}\right)[/tex].

We can apply some of are rules:

[tex]\sqrt{5}\sqrt{5}-\sqrt{5}\sqrt{3}+\sqrt{3}\sqrt{5}-\sqrt{3}\sqrt{3}-\sqrt{15}\sqrt{5}+\sqrt{15}\sqrt{3}[/tex]

And, now to simplifying!

Adding similar elements:

[tex]\sqrt{5}\sqrt{5}-\sqrt{3}\sqrt{3}-\sqrt{15}\sqrt{5}+\sqrt{15}\sqrt{3}[/tex]

Simplifying and cleaning up each radical:

[tex]5-3-5\sqrt{3}+3\sqrt{5}[/tex].

Now all we have to do is add the similar elements (again.)

In doing so we reach an answer of [tex]\boxed{2+3\sqrt{5}}[/tex]

Hope this helps!

- Sahkfam

ACCESS MORE