If `a^(2) + (1)/(a^(2))= 23 and a ne 0`, find the value of `a^(3) + (1)/(a^(3))`

Answer

To solve the problem, we will follow these steps:

### Step 1: Given Information
We start with the equation:
\[ a^2 + \frac{1}{a^2} = 23 \]

### Step 2: Find \(a + \frac{1}{a}\)
We know that:
\[ \left(a + \frac{1}{a}\right)^2 = a^2 + 2 + \frac{1}{a^2} \]
Thus, we can rearrange this to find \(a + \frac{1}{a}\):
\[ \left(a + \frac{1}{a}\right)^2 = a^2 + \frac{1}{a^2} + 2 \]
Substituting the known value:
\[ \left(a + \frac{1}{a}\right)^2 = 23 + 2 = 25 \]
Taking the square root:
\[ a + \frac{1}{a} = \sqrt{25} = 5 \]

### Step 3: Find \(a^3 + \frac{1}{a^3}\)
We will use the identity:
\[ a^3 + \frac{1}{a^3} = \left(a + \frac{1}{a}\right)^3 - 3\left(a + \frac{1}{a}\right) \]
Substituting \(a + \frac{1}{a} = 5\):
\[ a^3 + \frac{1}{a^3} = 5^3 - 3 \times 5 \]
Calculating \(5^3\):
\[ 5^3 = 125 \]
Calculating \(3 \times 5\):
\[ 3 \times 5 = 15 \]
Now substituting these values back:
\[ a^3 + \frac{1}{a^3} = 125 - 15 = 110 \]

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

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