a^1/3 +b^1/3 +c^1/3=(2^1/3 -1)^1/3 find 3(a+b+c)

Let's solve the given problem step by step.

### Problem:
\[
a^{1/3} + b^{1/3} + c^{1/3} = (2^{1/3} - 1)^{1/3}
\]
We need to find \( 3(a + b + c) \).

### Solution:
1. Let \( x = a^{1/3} \), \( y = b^{1/3} \), and \( z = c^{1/3} \). The equation can be rewritten as:
   \[
   x + y + z = (2^{1/3} - 1)^{1/3}
   \]

2. Cubing both sides to eliminate the cube roots:
   \[
   (x + y + z)^3 = \left((2^{1/3} - 1)^{1/3}\right)^3
   \]
   Simplifying the right side gives:
   \[
   (x + y + z)^3 = 2^{1/3} - 1
   \]

3. Expanding the left-hand side:
   \[
   x^3 + y^3 + z^3 + 3(xy + yz + zx)(x + y + z) = 2^{1/3} - 1
   \]
   Since \( x^3 = a \), \( y^3 = b \), and \( z^3 = c \), we have:
   \[
   a + b + c + 3(xy + yz + zx)(x + y + z) = 2^{1/3} - 1
   \]

   This expression simplifies if we can identify \( xy + yz + zx \) and \( x + y + z \). However, we need to find \( a + b + c \) from this.

4. Now, let's try to assume symmetric values for \( x \), \( y \), and \( z \). If we find a solution where \( x = y = z \), then we can proceed further. Let \( x = y = z \), which implies:
   \[
   3x = (2^{1/3} - 1)^{1/3}
   \]

5. Solving for \( x \):
   \[
   x = \frac{(2^{1/3} - 1)^{1/3}}{3}
   \]

6. Substituting back, we get:
   \[
   a + b + c = 3x^3
   \]

Thus, to find \( 3(a + b + c) \), we have:
\[
3(a + b + c) = 9x^3
\]

Using our assumption, \( x = (2^{1/3} - 1)^{1/3} / 3 \). 

Hence:
\[
3(a + b + c) = 9 \cdot \left( \frac{(2^{1/3} - 1)^{1/3}}{3} \right)^3 = (2^{1/3} - 1)
\]

So, the final answer is:
\[
\boxed{3(a + b + c) = 2^{1/3} - 1}
\]

Would you like more details or have any other questions?

### Related Questions:
1. How do we solve equations involving cube roots effectively?
2. What are the steps for expanding expressions of the form \((x + y + z)^3\)?
3. Can we generalize this method to higher powers (e.g., fourth roots)?
4. What are some properties of cube roots that simplify equations?
5. How do we check if a cubic equation has a unique real solution?

**Tip:** When dealing with roots of higher degree (like cube roots), it can help to try symmetric substitutions or specific values to simplify the problem.

a^1/3 + b^1/3 + c^1/3 = (2^1/3 - 1)^1/3 find 3(a + b + c)

Discover how to solve the cube root equation a^1/3 + b^1/3 + c^1/3 = (2^1/3 - 1)^1/3 and find 3(a + b + c). This solution covers cube roots, symmetric substitution, and expansion formulas, ideal for advanced algebra students in Grades 10-12.

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