Prove that (x+1/x)(x^2+1/x^2)(x^4+1/x^4)(x^8+1/x^8) = x^32-1 / x^15(x^2-1)

Question

Prove that (x+1/x)(x^2+1/x^2)(x^4+1/x^4)(x^8+1/x^8) = x^32-1 / x^15(x^2-1)

Answer 1

It has to be x + (1/x), because it isn't true with (x+1)/x, so we couldn't prove it. Because of order of operations, the way you have written it is correct.

 

The first thing to do is get the expressions in each of your parentheses to be a fraction, instead of two terms being added. This gives us

 

[(x²+1)/x][(x⁴+1)/x⁴][(x⁸+1)/x⁸][(x¹⁶+1)/x¹⁶]

 

Since multiplying of fractions is done straight across, this give us

 

(x²+1)(x⁴+1)(x⁸+1)(x¹⁶+1)/x¹⁵

 

Now it's starting to look a little like the right side, in that both sides have x¹⁵ in the denominator.

 

Observe that the right side also has (x²-1) in the denominator. Since the left side doesn't, let's multiply the denominator on the left side by (x²-1). Of course, we'll do it to the numerator, too.

 

Now the left side is  [(x²-1)(x²+1)(x⁴+1)(x⁸+1)(x¹⁶+1)]/[x¹⁵(x²-1)]  whose denominator looks just like the right side. This is a good sign that we're headed in the right direction.

 

Now we do some multiplying in the numerator.

 

(x²-1)(x²+1) = x⁴-1, so we now have   (x⁴-1)(x⁴+1)(x⁸+1)(x¹⁶+1)/denominator   (I don't feel like entering the denom each time I do a step.

 

Do another multiplication:   (x⁴-1)(x⁴+1) = x⁸-1.  You should be able to see that you will multiply that x⁸-1 by the x⁸+1 and get x¹⁶-1, which you will then multiply by the only thing left in the top, which is  x¹⁶+1, finally getting us

 

 

(x³²-1)/[x¹⁵(x²-1)]  which is the right side of the equation.

 

Whew!!!!!

 

Another option would be to factor the  x³²-1 on the right side and reduce the fraction.