How does sqrt{x^[18] + x^[10]} equal (x^5)*sqrt{x^8 + 1} ?

blakebranham5

New member
How does this equation equal E?

MarkFL

Super Moderator
Staff member
Hello, and welcome to FMH!

What you have is an expression, not an equation.

$$\displaystyle \sqrt{x^{18}+x^{10}}=\sqrt{x^{10}\left(x^{8}+1\right)}=x^5\sqrt{x^{8}+1}$$

Does that make sense?

ksdhart2

Senior Member
Hint: $$\displaystyle x^{18} = x^{10} \cdot x^8$$, so $$\displaystyle x^{18} + x^{10} = x^{10}(x^8 + 1)$$.

blakebranham5

New member
Thank you. I also need help with the following 2 questions.

Moderator Note: Two images removed; reposted as separate threads.

tkhunny

Moderator
Staff member
In the post #1 question, where did you use "x is a positive number"?

Staff member

blakebranham5

New member
Thank you I will do that! I’m just trying to learn how to solve these types of questions so I will know how to do them on an upcoming exam.

tkhunny

Moderator
Staff member
Thank you I will do that! I’m just trying to learn how to solve these types of questions so I will know how to do them on an upcoming exam.

MarkFL

Super Moderator
Staff member
It's a good question too, because understand that will help a student avoid many pitfalls down the road.

blakebranham5

New member
Sorry just seen the question. Im not positive where X is used as a positive Integer. I’m still not certain on how a problem like this is done.

MarkFL

Super Moderator
Staff member
Sorry just seen the question. Im not positive where X is used as a positive Integer. I’m still not certain on how a problem like this is done.
If we don't know the sign of $$x$$, then we must write:

$$\displaystyle \sqrt{x^{2n}}=\left|x^n\right|$$

For example:

$$\displaystyle \sqrt{(-1)^2}\ne-1$$

$$\displaystyle \sqrt{(-1)^2}=|-1|=1$$
So, in the given problem, knowing that $$x$$ is positive, allows us to write:
$$\displaystyle \sqrt{x^{10}}=x^5$$