Not sure how to solve this question

[MadaMars]

Find the numeric value of (p/q)^3 given that 1/(p+q) = 1/p + 1/q

 

[MarkFL]

Hello, and welcome to FMH!

Rewrite the equation as:

[MATH](p+q)^2=pq[/MATH]
Divide by $q^2$:

[MATH]\left(\frac{p}{q}+1\right)^2=\frac{p}{q}[/MATH]
Expand and collect like terms:

[MATH]\left(\frac{p}{q}\right)^2+\frac{p}{q}+1=0[/MATH]
Multiply by $\frac{p}{q}$:

[MATH]\left(\frac{p}{q}\right)^3+\left(\frac{p}{q}\right)^2+\frac{p}{q}=0[/MATH]
What do the two previous equations then imply?

 

[MarkFL]

To follow up, if we subtract the former equation from the latter, we obtain:

[MATH]\left(\frac{p}{q}\right)^3-1=0[/MATH]
Thus:

[MATH]\left(\frac{p}{q}\right)^3=1[/MATH]
Another approach would be to begin at the same starting point:

[MATH](p+q)^2=pq[/MATH]
Expand and collect like terms:

[MATH]p^2+pq+q^2=0[/MATH]
We should recognize this could be a factor in a difference of cubes. Observing that in the original equation $p\ne q$ we may multiply by $p-q$ to get:

[MATH](p-q)(p^2+pq+q^2)=0[/MATH]
And so:

[MATH]p^3-q^3=0[/MATH]
Divide by $q^3$:

[MATH]\left(\frac{p}{q}\right)^3-1=0[/MATH]
Thus:

[MATH]\left(\frac{p}{q}\right)^3=1[/MATH]