Simplify the radical expression: the fifth root of 729x⁶ divided by the fifth root of 3x.

Step-by-step solution

1. 1

   Merge the two radicals. Radicals with the same index divide term by term: $\dfrac{\sqrt[5]{A}}{\sqrt[5]{B}} = \sqrt[5]{A/B}$. Doing this first turns an awkward quotient into one simple radicand.

2. 2

   Reduce the radicand. $\dfrac{729x^6}{3x} = 243x^5$, since $729 \div 3 = 243$ and $x^{6-1} = x^5$.

3. 3

   Recognise the perfect fifth power. $243 = 3^5$, and $x^5$ is already a fifth power, so $243x^5 = (3x)^5$.

4. 4

   Take the fifth root. $\sqrt[5]{(3x)^5} = 3x$. A fifth root is an odd root, so no absolute-value bars are needed.

5. 5

   Check numerically. At $x = 2$: $\sqrt[5]{729 \cdot 64} = \sqrt[5]{46656} \approx 8.58581$ and $\sqrt[5]{6} \approx 1.43097$; their ratio is $6.00000 = 3 \cdot 2$.