The expression `((log_R((a)/b))^(3)+(log_(R)((b)/(c)))^(3)+(log _(R)((c)/(a)))^(3))/((log_(R)((a)/(b)))(log_(R)((b)/(c)))(log_(R)((c)/(a))))(a,b,c,Rgt0andRne1)` wherever defined, simplifies to

Answer

To simplify the given expression

\[ \frac{(\log_R(\frac{a}{b}))^3 + (\log_R(\frac{b}{c}))^3 + (\log_R(\frac{c}{a}))^3}{\log_R(\frac{a}{b}) \cdot \log_R(\frac{b}{c}) \cdot \log_R(\frac{c}{a})} \]

we will follow these steps:

### Step 1: Define Variables
Let:
- \( x = \log_R\left(\frac{a}{b}\right) \)
- \( y = \log_R\left(\frac{b}{c}\right) \)
- \( z = \log_R\left(\frac{c}{a}\right) \)

### Step 2: Rewrite the Expression
The expression can now be rewritten as:

\[ \frac{x^3 + y^3 + z^3}{xyz} \]

### Step 3: Use the Identity for Cubes
We can use the identity for the sum of cubes, which states that:

\[ x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx) \]

### Step 4: Check if \( x + y + z = 0 \)
We need to check whether \( x + y + z = 0 \):

\[ x + y + z = \log_R\left(\frac{a}{b}\right) + \log_R\left(\frac{b}{c}\right) + \log_R\left(\frac{c}{a}\right) \]

Using the properties of logarithms, we can combine these:

\[ x + y + z = \log_R\left(\frac{a}{b} \cdot \frac{b}{c} \cdot \frac{c}{a}\right) = \log_R(1) = 0 \]

### Step 5: Apply the Identity
Since \( x + y + z = 0 \), we can apply the identity:

\[ x^3 + y^3 + z^3 = 3xyz \]

### Step 6: Substitute Back into the Expression
Now substituting back into our expression:

\[ \frac{3xyz}{xyz} = 3 \]

### Final Result
Thus, the simplified expression is:

\[ \boxed{3} \]