To find the edge length of the cube, we need to use the formula for the volume of a cube:

$V = s^3$

Where $V$ is the volume and $s$ is the edge length of the cube. Given that the volume $V = 7290 \, \text{cm}^3$, we can solve for $s$:

$s^3 = 7290$

Now, to find $s$, we take the cube root of both sides:

$s = \sqrt[3]{7290}$

We simplify this expression by factoring 7290 into its prime factors:

$7290 = 2 \times 3^6 \times 5$

So,

[ s = \sqrt[3]{2 \times 3^6 \times 5} = 3^2 \times \sqrt[3]{2 \times 5} = 9 \sqrt[3]{10} ]

Thus, the edge length of the cube is:

$s = 9 \sqrt[3]{10} \, \text{cm}$

So the correct answer is option b: $9 \sqrt[3]{10} \, \text{cm}$.

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Would you like more details or clarification on any step?

Here are five related questions:

1. How can you simplify cube roots for other numbers?
2. What is the relationship between volume and surface area for cubes?
3. How do you solve for cube roots manually without a calculator?
4. Can the cube root of a non-perfect cube ever be a whole number?
5. What is the significance of prime factorization in simplifying cube roots?

Tip: Remember to always break down larger numbers into prime factors when dealing with roots to simplify expressions easily!