If a + 1/a = 3 1/3 find the value of a ^ 3 - 1/(a ^ 3)

We are given the equation:

\[
a + \frac{1}{a} = 3 \frac{1}{3} = \frac{10}{3}
\]

We need to find the value of:

\[
a^3 - \frac{1}{a^3}
\]

### Step 1: Let \( a + \frac{1}{a} = x \)
Given that \( x = \frac{10}{3} \), we now need to express \( a^3 - \frac{1}{a^3} \) in terms of \( x \).

### Step 2: Use the identity for cubes
The identity that relates \( a^3 - \frac{1}{a^3} \) to \( a + \frac{1}{a} \) is:

\[
a^3 - \frac{1}{a^3} = \left( a + \frac{1}{a} \right) \left( a^2 - 1 + \frac{1}{a^2} \right)
\]

First, we need to find the value of \( a^2 + \frac{1}{a^2} \). We can use the square of \( a + \frac{1}{a} \):

\[
\left( a + \frac{1}{a} \right)^2 = a^2 + 2 + \frac{1}{a^2}
\]

So,

\[
\left( \frac{10}{3} \right)^2 = a^2 + 2 + \frac{1}{a^2}
\]

\[
\frac{100}{9} = a^2 + 2 + \frac{1}{a^2}
\]

Subtract 2 from both sides:

\[
a^2 + \frac{1}{a^2} = \frac{100}{9} - 2 = \frac{100}{9} - \frac{18}{9} = \frac{82}{9}
\]

### Step 3: Find \( a^3 - \frac{1}{a^3} \)
Now, use the identity:

\[
a^3 - \frac{1}{a^3} = \left( a + \frac{1}{a} \right) \left( a^2 - 1 + \frac{1}{a^2} \right)
\]

Simplifying this, we find:

\[
a^3 - \frac{1}{a^3} = \left( \frac{10}{3} \right) \left( \frac{82}{9} \right)
\]

Multiply these fractions:

\[
a^3 - \frac{1}{a^3} = \frac{10 \times 82}{3 \times 9} = \frac{820}{27}
\]

Thus, the value of \( a^3 - \frac{1}{a^3} \) is:

\[
\boxed{\frac{820}{27}}
\]

Would you like more details, or do you have any questions?

Here are some related questions to explore:
1. How does this formula change if \( a + \frac{1}{a} \) equals a different value?
2. Can you derive similar identities for higher powers of \( a \)?
3. What if \( a + \frac{1}{a} \) were a complex number?
4. How can you use the same approach for negative exponents?
5. Can you derive a general formula for \( a^n - \frac{1}{a^n} \)?

**Tip**: When solving equations with fractions, always consider simplifying expressions using identities for sums, products, and powers.

Discover how to find the value of a^3 - 1/(a^3) given the equation a + 1/a = 3 1/3. Follow a detailed step-by-step solution using algebraic identities and cube root formulas. Suitable for advanced high school students.

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