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The problem is as follows:

We are given positive real numbers \( a \), \( b \), and \( c \), with the condition:

\[
\frac{1}{a^2} + \frac{8}{b^2} + \frac{1}{c^2} = 1
\]

We are asked to find the minimum value of \( a + b + c \).

### Step-by-Step Solution

#### Step 1: Recognizing the inequality
This is a typical minimization problem under a constraint, and we can apply techniques such as the **method of Lagrange multipliers** or **Cauchy-Schwarz inequality**.

We will use the **Cauchy-Schwarz inequality** to estimate the minimum value of \( a + b + c \).

#### Step 2: Applying Cauchy-Schwarz inequality
The Cauchy-Schwarz inequality in the context of sums of squares states:

\[
\left( \frac{1^2}{a^2} + \frac{8^2}{b^2} + \frac{1^2}{c^2} \right) \left( a^2 + b^2 + c^2 \right) \geq (1 + 8 + 1)^2 = 10^2 = 100
\]

Given that:

\[
\frac{1}{a^2} + \frac{8}{b^2} + \frac{1}{c^2} = 1
\]

We substitute into the inequality:

\[
(1)(a^2 + b^2 + c^2) \geq 100
\]

Thus, we have:

\[
a^2 + b^2 + c^2 \geq 100
\]

#### Step 3: Estimating \( a + b + c \)
Now, we estimate the minimum value of \( a + b + c \). Since:

\[
a^2 + b^2 + c^2 \geq 100
\]

and \( a + b + c \geq \sqrt{a^2 + b^2 + c^2} \), we find:

\[
a + b + c \geq \sqrt{100} = 10
\]

Thus, the minimum possible value of \( a + b + c \) is **10**.

### Conclusion
The minimum value of \( a + b + c \) is \( \boxed{10} \).

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

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Here are 5 related questions:

1. How does the Cauchy-Schwarz inequality help in optimization problems?
2. Can we use Lagrange multipliers to solve this problem differently?
3. What is the difference between \( a^2 + b^2 + c^2 \) and \( a + b + c \)?
4. How can we verify that the minimum value of \( a + b + c \) is achieved?
5. What are some other inequality techniques used for such problems?

**Tip:** Inequalities like Cauchy-Schwarz or AM-GM are powerful tools for optimization problems in mathematics, especially when dealing with sums of variables under constraints.

已知 a,b,c ∈ R⁺ 且 1/a² + 8/b² + 1/c² = 1，求 a + b + c 得最小值。

Learn how to find the minimum value of a + b + c when constrained by the equation 1/a² + 8/b² + 1/c² = 1. This problem uses optimization techniques like the Cauchy-Schwarz inequality. Suitable for advanced high school or college-level students.

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