Simplifying Expression with Positive Variables a, b, and c | Step-by-Step Guide

[BruceSpringste]

Homework Statement

Om a > 0, b > 0, c > 0, och x = ((ab√c)^1/3-a(b²c)^1/4)/(a³b²c)^1/6

The Attempt at a Solution

I have no idea where to start. I understand the relevance of a,b and c being > 0 in order to simplify but other than that I am pretty much stuck!

 

[HallsofIvy]

A first obvious step is to replace that $\sqrt{c}$ with $c^{1/2}$. Then use the "laws of exponentials: $(abc^{1/2})^{1/3}= a^{1/3}b^{1/3}c^{1/6}$, $a(b^2c)^{1/4}= ab^{1/2}c^{1/4}$ and $(a^3b^2c)^{1/6}= a^{1/2}b^{1/3}c^{1/6}$

So you have $$\frac{a^{1/3}b^{1/3}c^{1/6}- ab^{1/2}c^{1/4}}{a^{1/2}b^{1/3}c^{1/6}}$$

Factor the largest power of a, b, and c in both terms in the numerator and cancel what you can with the denominator.

 

[BruceSpringste]

\begin{align*}
\frac{\sqrt[3]{ab \sqrt{c}} - a \sqrt[4]{b^2 c}}{\sqrt[6]{a^3 b^2 c}}
\end{align*}

Wrote the equation in latex too, it was easier to see.

 

[BruceSpringste]

Alright that was easier than I previously thought! Thanks!