Answer 1

Cube root by factors:
On factorising 74088000 into prime factors, we get:

\[74088000 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 5 \times 5 \times 5 \times 7 \times 7 \times 7\

On grouping the factors in triples of equal factors, we get:

\[74088000 = \left\{ 2 \times 2 \times 2 \right\} \times \left\{ 2 \times 2 \times 2 \right\} \times \left\{ 3 \times 3 \times 3 \right\} \times \left\{ 5 \times 5 \times 5 \right\} \times \left\{ 7 \times 7 \times 7 \right\}\]

Now, taking one factor from each triple, we get:

\[\sqrt[3]{74088000} = 2 \times 2 \times 3 \times 5 \times 7 = 420\]