MHB Solve Exponential Function: x1/2 + x-1/2 | Help & Explanation

[linapril]

Hello! Could someone please help me answer this question, and explain how you answered it?

1. (x^1/2 + x^-1/2)²​Thank you!

 

[Ackbach]

What are you asked to do? And what steps have you made towards doing that?

 

[linapril]

It just says to simplify it.
I multiplied away the brackets, getting x + x, which equals 2x. But this answer's wrong apparently...

 

[Ackbach]

Yes, there are several errors in your computations. The first error, seemingly, is that you distributed the exponent over addition. This NEVER WORKS! If you don't believe me, check this out: is
$$25=5^{2}=(2+3)^{2}=2^{2}+3^{2}=4+9=13?$$
Definitely not. So this thought doesn't work. It's so common it's called the "freshman's dream". You have to write out your expression and FOIL it:
$$(x^{1/2}+x^{-1/2})^{2}=(x^{1/2}+x^{-1/2})(x^{1/2}+x^{-1/2}).$$
Now do your firsts, outers, inners, and lasts. What do you get?

 

[linapril]

I got the right answer, thank you so much :)

 

[Ackbach]

You're welcome! Have a good one.