Find : `a^3 + (1)/( a^3)`, if `a + (1)/( a) =5`.

Answer

To find \( a^3 + \frac{1}{a^3} \) given that \( a + \frac{1}{a} = 5 \), we can use the identity for the cube of a sum. Here are the steps:

### Step 1: Use the identity for cubes
We know that:
\[ \left( a + \frac{1}{a} \right)^3 = a^3 + \frac{1}{a^3} + 3 \left( a \cdot \frac{1}{a} \right) \left( a + \frac{1}{a} \right) \]
This simplifies to:
\[ \left( a + \frac{1}{a} \right)^3 = a^3 + \frac{1}{a^3} + 3 \left( 1 \right) \left( a + \frac{1}{a} \right) \]
Thus:
\[ \left( a + \frac{1}{a} \right)^3 = a^3 + \frac{1}{a^3} + 3 \left( a + \frac{1}{a} \right) \]

### Step 2: Substitute the known value
From the problem, we know that:
\[ a + \frac{1}{a} = 5 \]
Now, substituting this value into the equation:
\[ 5^3 = a^3 + \frac{1}{a^3} + 3 \cdot 5 \]

### Step 3: Calculate \( 5^3 \)
Calculating \( 5^3 \):
\[ 5^3 = 125 \]
So we have:
\[ 125 = a^3 + \frac{1}{a^3} + 15 \]

### Step 4: Solve for \( a^3 + \frac{1}{a^3} \)
Now, we can isolate \( a^3 + \frac{1}{a^3} \):
\[ a^3 + \frac{1}{a^3} = 125 - 15 \]
\[ a^3 + \frac{1}{a^3} = 110 \]

### Final Answer
Thus, the value of \( a^3 + \frac{1}{a^3} \) is:
\[ \boxed{110} \]