0 equals square root of 64 minus x^3 solve for x

The equation you're trying to solve is:

\[
0 = \sqrt{64 - x^3}
\]

### Step 1: Square both sides of the equation to eliminate the square root.
Since \(\sqrt{64 - x^3} = 0\), squaring both sides gives:

\[
0^2 = (\sqrt{64 - x^3})^2
\]
\[
0 = 64 - x^3
\]

### Step 2: Solve for \(x^3\).
Now, isolate \(x^3\):

\[
x^3 = 64
\]

### Step 3: Take the cube root of both sides.
To find \(x\), take the cube root of both sides:

\[
x = \sqrt[3]{64}
\]
\[
x = 4
\]

### Solution:
The solution to the equation is:

\[
x = 4
\]

Let me know if you'd like a detailed explanation or have any questions!

Here are 5 related questions to expand your understanding:
1. How would the equation change if it were \(0 = \sqrt{64 - x^2}\)?
2. What are the possible solutions for \(x^3 = -64\)?
3. How would you solve \(0 = \sqrt{81 - x^4}\)?
4. How does taking a square root differ from taking a cube root in terms of solutions?
5. What is the difference between real and complex solutions for cube roots?

**Tip:** When solving equations involving roots, always check for extraneous solutions by substituting back into the original equation.

Learn how to solve the equation 0 = sqrt(64 - x^3) step by step. This problem involves key algebraic concepts like squaring both sides and solving cubic equations using cube roots. Suitable for students in Grades 9-11.

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