We are given the equation:

\[
(X - 6)^4 + (X - 8)^4 = 16
\]

### Step 1: Introduce a substitution
To simplify the problem, let's introduce a substitution:

\[
y = X - 7
\]

Thus, we have:

\[
X - 6 = y + 1 \quad \text{and} \quad X - 8 = y - 1
\]

Substituting these into the original equation:

\[
(y + 1)^4 + (y - 1)^4 = 16
\]

### Step 2: Expand the binomials
Now let's expand \((y + 1)^4\) and \((y - 1)^4\).

For \((y + 1)^4\):

\[
(y + 1)^4 = y^4 + 4y^3 + 6y^2 + 4y + 1
\]

For \((y - 1)^4\):

\[
(y - 1)^4 = y^4 - 4y^3 + 6y^2 - 4y + 1
\]

### Step 3: Add the two expressions
Now add \((y + 1)^4\) and \((y - 1)^4\):

\[
(y^4 + 4y^3 + 6y^2 + 4y + 1) + (y^4 - 4y^3 + 6y^2 - 4y + 1)
\]

Simplifying this:

\[
2y^4 + 12y^2 + 2 = 16
\]

### Step 4: Solve the equation
We can now simplify this equation:

\[
2y^4 + 12y^2 + 2 = 16
\]

Subtract 16 from both sides:

\[
2y^4 + 12y^2 - 14 = 0
\]

Divide through by 2:

\[
y^4 + 6y^2 - 7 = 0
\]

Now let \( z = y^2 \). So the equation becomes:

\[
z^2 + 6z - 7 = 0
\]

### Step 5: Solve the quadratic equation
We can solve this quadratic equation using the quadratic formula:

\[
z = \frac{-6 \pm \sqrt{6^2 - 4(1)(-7)}}{2(1)} = \frac{-6 \pm \sqrt{36 + 28}}{2} = \frac{-6 \pm \sqrt{64}}{2}
\]

\[
z = \frac{-6 \pm 8}{2}
\]

So, we have two solutions for \(z\):

\[
z = \frac{-6 + 8}{2} = 1 \quad \text{or} \quad z = \frac{-6 - 8}{2} = -7
\]

Since \(z = y^2\), and \(y^2\) cannot be negative, we discard \(z = -7\).

Thus, \(y^2 = 1\), which gives \(y = \pm 1\).

### Step 6: Find the value of \(X\)
Recall that \(y = X - 7\), so:

- If \(y = 1\), then \(X - 7 = 1\), which gives \(X = 8\).
- If \(y = -1\), then \(X - 7 = -1\), which gives \(X = 6\).

### Final Answer:
The solutions are \(X = 6\) and \(X = 8\).

Would you like more details on any step? 

Here are 5 related questions:
1. What other techniques can be used to solve fourth-degree equations?
2. How can the binomial theorem be applied in expanding polynomial expressions?
3. What happens if we don't use substitution? Is there a direct way?
4. Can we solve equations with higher powers using similar methods?
5. How does the quadratic formula handle complex numbers?

**Tip**: Always look for symmetry in equations; it often helps simplify the problem!

Learn how to solve the equation (X - 6)^4 + (X - 8)^4 = 16 step-by-step. This problem involves fourth-degree polynomial equations, substitution techniques, and binomial expansion. Suitable for advanced high school students and college-level mathematics enthusiasts.

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