We are tasked with simplifying the following expression:

$6(9000)^{1/3} + 8(576)^{1/3}$

Step 1: Simplifying the cube roots

First, let's evaluate the cube roots:

1. $(9000)^{1/3}$ is the cube root of 9000. This simplifies to approximately: $(9000)^{1/3} \approx 20.88$

2. $(576)^{1/3}$ is the cube root of 576. This simplifies to approximately: $(576)^{1/3} \approx 8.32$

Step 2: Multiply by the coefficients

Next, multiply these cube roots by their respective coefficients:

1. $6 \times (9000)^{1/3} \approx 6 \times 20.88 = 125.28$

2. $8 \times (576)^{1/3} \approx 8 \times 8.32 = 66.56$

Step 3: Add the results

Now, add the two results together:

$125.28 + 66.56 = 191.84$

Thus, the simplified value of the expression is approximately:

$\boxed{191.84}$

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Would you like further details on any part of the calculation?

Here are some related questions to explore further:

1. What are cube roots, and how do you calculate them exactly?
2. How can we estimate cube roots for larger numbers more efficiently?
3. What happens if we use an exact solution instead of an approximation?
4. How would this expression change if the exponents were different (e.g., square roots instead of cube roots)?
5. How can logarithms be used to compute roots?

Tip: When working with cube roots, understanding prime factorization can sometimes simplify the calculations significantly.