Simplify: \(\frac{\sqrt[3]{5832}}{\sqrt[4]{1296}} \times \frac{3}{54} \times…

Concept: An n-th root is evaluated by writing the radicand in prime-factorised form and dividing each exponent by n, because \(\sqrt[n]{a^n} = a\). Once every radical has collapsed to a whole number, what remains is an ordinary product of fractions, in which common factors are cancelled before multiplying.

Application to this expression:

1. Prime-factorise the first radicand: \(5832 = 2^3 \times 3^6\), so \(\sqrt[3]{5832} = 2^{3/3} \times 3^{6/3} = 2 \times 9 = 18\).

2. Prime-factorise the second radicand: \(1296 = 2^4 \times 3^4\), so \(\sqrt[4]{1296} = 2 \times 3 = 6\).

3. The radical part therefore reduces to \(\frac{18}{6} = 3\).

4. Reduce the remaining fraction: \(\frac{3}{54} = \frac{1}{18}\).

5. Multiply what is left: \(3 \times \frac{1}{18} \times 198 = \frac{198}{6} = 33\).

Cross-check: \(18^3 = 5832\) and \(6^4 = 1296\) confirm both roots, and evaluating the whole product in a single line gives \(\frac{18 \times 3 \times 198}{6 \times 54} = \frac{10692}{324} = 33\).