Understanding the Problem: Finding \(a^3 + \frac{1}{a^3}\)

The question asks us to find the value of the expression \(a^3 + \frac{1}{a^3}\), given that we know the value of \(a + \frac{1}{a}\). This is a common type of algebraic problem that can be solved using algebraic identities.

We are given:

• \(a + \frac{1}{a} = 5\)

We need to find the value of:

• \(a^3 + \frac{1}{a^3}\)

Using Algebraic Identities to Solve for \(a^3 + \frac{1}{a^3}\)

To find the value of \(a^3 + \frac{1}{a^3}\), we can use the algebraic identity for the cube of a sum:

The identity is: \((x+y)^3 = x^3 + y^3 + 3xy(x+y)\)

Let's apply this identity by setting \(x = a\) and \(y = \frac{1}{a}\). When we do this, the term \(xy\) becomes \(a \times \frac{1}{a}\), which simplifies to 1.

Substituting \(x=a\) and \(y=\frac{1}{a}\) into the identity, we get:

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

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

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

Calculating the Value of \(a^3 + \frac{1}{a^3}\)

Now we can use the given information that \(a + \frac{1}{a} = 5\) and substitute this value into the derived equation.

\((5)^3 = a^3 + \frac{1}{a^3} + 3 (5)\)

Calculate the values:

• \(5^3 = 5 \times 5 \times 5 = 25 \times 5 = 125\)
• \(3 \times 5 = 15\)

Substitute these values back into the equation:

\(125 = a^3 + \frac{1}{a^3} + 15\)

Now, we need to isolate \(a^3 + \frac{1}{a^3}\). We can do this by subtracting 15 from both sides of the equation:

\(125 - 15 = a^3 + \frac{1}{a^3}\)

\(110 = a^3 + \frac{1}{a^3}\)

So, the value of \(a^3 + \frac{1}{a^3}\) is 110.

Summary of Steps to find \(a^3 + \frac{1}{a^3}\)

Here's a quick recap of the steps:

1. Start with the given equation: \(a + \frac{1}{a} = 5\).
2. Recognize the need for an identity involving cubes.
3. Use the identity \((x+y)^3 = x^3 + y^3 + 3xy(x+y)\).
4. Substitute \(x=a\) and \(y=\frac{1}{a}\) into the identity.
5. Simplify the expression to get \(\left(a + \frac{1}{a}\right)^3 = a^3 + \frac{1}{a^3} + 3 \left(a + \frac{1}{a}\right)\).
6. Substitute the given value \(a + \frac{1}{a} = 5\) into the simplified equation.
7. Solve for \(a^3 + \frac{1}{a^3}\).

Following these steps leads to the result \(a^3 + \frac{1}{a^3} = 110\).

Revision Table - Algebra Identities

Additional Information - Solving \(a^2 + \frac{1}{a^2}\)

Similarly, if you were asked to find \(a^2 + \frac{1}{a^2}\) given \(a + \frac{1}{a} = 5\), you would use the identity for the square of a sum:

\((x+y)^2 = x^2 + y^2 + 2xy\)

Setting \(x=a\) and \(y=\frac{1}{a}\), we get:

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

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

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

Substitute \(a + \frac{1}{a} = 5\):

\((5)^2 = a^2 + \frac{1}{a^2} + 2\)

\(25 = a^2 + \frac{1}{a^2} + 2\)

Solve for \(a^2 + \frac{1}{a^2}\):

\(25 - 2 = a^2 + \frac{1}{a^2}\)

\(23 = a^2 + \frac{1}{a^2}\)

This shows how different identities are used depending on the power required.