Finding the Value of a³ + 1/a³

This problem requires us to find the value of an algebraic expression given another related expression. We are given the value of \(a + \frac{1}{a}\) and asked to find the value of \(a^3 + \frac{1}{a^3}\).

We are given:

• \(a + \frac{1}{a} = 3\)

We need to find the value of \(a^3 + \frac{1}{a^3}\).

Using Algebraic Identities to Solve

To solve this, we can use the algebraic identity for the cube of a sum: \((x+y)^3 = x^3 + y^3 + 3xy(x+y)\).

Let's apply this identity by setting \(x = a\) and \(y = \frac{1}{a}\). So, we have:

\(\left(a + \frac{1}{a}\right)^3 = a^3 + \left(\frac{1}{a}\right)^3 + 3 \cdot a \cdot \frac{1}{a} \cdot \left(a + \frac{1}{a}\right)\)

Let's simplify the expression:

\(\left(a + \frac{1}{a}\right)^3 = a^3 + \frac{1}{a^3} + 3 \cdot 1 \cdot \left(a + \frac{1}{a}\right)\)

\(\left(a + \frac{1}{a}\right)^3 = a^3 + \frac{1}{a^3} + 3\left(a + \frac{1}{a}\right)\)

Calculation Steps

We know that \(a + \frac{1}{a} = 3\). Let's substitute this value into the equation we derived:

\((3)^3 = a^3 + \frac{1}{a^3} + 3(3)\)

Now, calculate the values:

\(27 = a^3 + \frac{1}{a^3} + 9\)

To find the value of \(a^3 + \frac{1}{a^3}\), we need to isolate it. Subtract 9 from both sides of the equation:

\(27 - 9 = a^3 + \frac{1}{a^3}\)

\(18 = a^3 + \frac{1}{a^3}\)

Conclusion

The value of \(a^3 + \frac{1}{a^3}\) is 18.

Revision Table: Key Concepts

Additional Information: Related Algebraic Identities

Understanding algebraic identities is crucial for solving many algebra problems. Here are a few other common identities:

• Square of a sum: \((x+y)^2 = x^2 + y^2 + 2xy\)
• Square of a difference: \((x-y)^2 = x^2 + y^2 - 2xy\)
• Cube of a difference: \((x-y)^3 = x^3 - y^3 - 3xy(x-y)\)
• Difference of squares: \(x^2 - y^2 = (x-y)(x+y)\)
• Sum of cubes: \(x^3 + y^3 = (x+y)(x^2 - xy + y^2)\)
• Difference of cubes: \(x^3 - y^3 = (x-y)(x^2 + xy + y^2)\)

These identities help simplify expressions and solve equations more efficiently, especially in problems involving powers of variables like the given question where we found the value of \(a^3 + \frac{1}{a^3}\) using the identity for \((a + \frac{1}{a})^3\).