To simplify $(9)^{\frac{1}{3}} + 3(576)^{\frac{1}{3}}$, follow these steps:

Step 1: Represent cube roots

The cube root of $9$ is $9^{\frac{1}{3}}$. Similarly, the cube root of $576$ is $576^{\frac{1}{3}}$.

Step 2: Approximate cube roots (if exact values are not possible)

For $9^{\frac{1}{3}}$: $9^{\frac{1}{3}} \approx 2.08008 \quad \text{(numerical approximation)}.$

For $576^{\frac{1}{3}}$: $576^{\frac{1}{3}} \approx 8.322 \quad \text{(numerical approximation)}.$

Step 3: Multiply and add

Now substitute these values into the expression: $9^{\frac{1}{3}} + 3 \cdot 576^{\frac{1}{3}} = 2.08008 + 3 \cdot 8.322$

Simplify further: $2.08008 + 24.966 = 27.04608.$

Final Answer:

$(9)^{\frac{1}{3}} + 3(576)^{\frac{1}{3}} \approx 27.05.$

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Would you like me to verify these calculations or provide further details on cube root approximations?

Related Questions:

1. What is the process for estimating cube roots of non-perfect cubes?
2. How do cube roots relate to exponents and their properties?
3. What are alternative methods to simplify expressions involving cube roots?
4. How would this calculation differ if $9$ and $576$ were perfect cubes?
5. What is the significance of cube roots in real-world applications like geometry?

Tip:

For perfect cube roots, check the factorization of the number to simplify the root exactly instead of approximating.