Solve the expression

[Albi]

Prove that the value of the expression does not depend on a, b, c and x
$$\frac{4}{a^{-x}+\frac{1}{b^{-x}+\frac{1}{c^{-x}}}} \div \frac{1}{a^{-x}+ \frac{1}{b^{-x}}}- \frac{4}{b^{-x}(a^{-x}b^{-x}c^{-x}+a^{-x}+ c^{-x})}$$
I have gotten up to this point:
$$\frac{4}{1+ \frac{a^xb^x}{1+ b^xc^x}}+ b^{x}- \frac{4}{\frac{1}{b^{x}}(\frac{1}{a^{x}b^{x}c^{x}}+\frac{1}{a^{x}}+\frac{1}{c^{x}})}$$
The solution in the book is 4, but I'm not sure that is correct, can someone help?

 

[Dr.Peterson]

Prove that the value of the expression does not depend on a, b, c and x
$$\frac{4}{a^{-x}+\frac{1}{b^{-x}+\frac{1}{c^{-x}}}} \div \frac{1}{a^{-x}+ \frac{1}{b^{-x}}}- \frac{4}{b^{-x}(a^{-x}b^{-x}c^{-x}+a^{-x}+ c^{-x})}$$
I have gotten up to this point:
$$\frac{4}{1+ \frac{a^xb^x}{1+ b^xc^x}}+ b^{x}- \frac{4}{\frac{1}{b^{x}}(\frac{1}{a^{x}b^{x}c^{x}}+\frac{1}{a^{x}}+\frac{1}{c^{x}})}$$
The solution in the book is 4, but I'm not sure that is correct, can someone help?

The answer is indeed 4. (Technically, you aren't solving the expression, but simplifying it, in this case expecting a constant.)

It looks like you broke a fraction apart into a sum, rather than pursuing a common denominator so you could make it a single fraction, as I would (eventually) do. You've also made a mistake somewhere, because your last line does not simplify to 4. (I used technology to check.)

I would start as you evidently did, trying to eliminate negative exponents, as well as simplifying complex fractions by multiplying by denominators. For example, my next step, if your stopping point were correct, would be to multiply numerator and denominator of the last fraction by $a^xb^xc^x$, and the first by $1+b^xc^x$.

Try again from the start, and show your steps (an image of work on paper is fine) so we can find errors or suggest different approaches. It's hard to judge from one wrong step what needs fixing.

 

[Albi]

The answer is indeed 4. (Technically, you aren't solving the expression, but simplifying it, in this case expecting a constant.)

It looks like you broke a fraction apart into a sum, rather than pursuing a common denominator so you could make it a single fraction, as I would (eventually) do. You've also made a mistake somewhere, because your last line does not simplify to 4. (I used technology to check.)

I would start as you evidently did, trying to eliminate negative exponents, as well as simplifying complex fractions by multiplying by denominators. For example, my next step, if your stopping point were correct, would be to multiply numerator and denominator of the last fraction by $a^xb^xc^x$, and the first by $1+b^xc^x$.

Try again from the start, and show your steps (an image of work on paper is fine) so we can find errors or suggest different approaches. It's hard to judge from one wrong step what needs fixing.

That's what I did up until this point, any suggestions what to do further, provided that I didn't make any mistake

 

[Dr.Peterson]

Good work. Now just do the same thing with the second fraction,



Specifically, multiply the numerator and denominator by the LCD, and everything soon falls into place.