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

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To simplify the expression (x1/2 + x-1/2)², it is crucial to avoid incorrectly distributing the exponent over addition, a common mistake known as the "freshman's dream." Instead, the expression should be expanded using the FOIL method, which involves multiplying the terms as (x1/2 + x-1/2)(x1/2 + x-1/2). This results in the correct simplification of the expression. The discussion highlights the importance of proper mathematical operations in solving exponential functions. Understanding these fundamentals is essential for accurate problem-solving in algebra.
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Hello! Could someone please help me answer this question, and explain how you answered it?

1. (x1/2 + x-1/2)2​Thank you!
 
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What are you asked to do? And what steps have you made towards doing that?
 
It just says to simplify it.
I multiplied away the brackets, getting x + x, which equals 2x. But this answer's wrong apparently...
 
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?
 
I got the right answer, thank you so much :)
 
You're welcome! Have a good one.
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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