Understanding the Problem: Finding x⁵ + 1/x⁵

We are given the equation \( \rm x + \frac{1}{x} = 4 \) and asked to find the value of \( \rm x^5 + \frac{1}{x^5} \).

This type of problem is common in algebra and requires using algebraic identities to find higher powers of \( \rm x + \frac{1}{x} \).

Step-by-Step Solution to Calculate x⁵ + 1/x⁵

To find \( \rm x^5 + \frac{1}{x^5} \), we can utilize the values of \( \rm x^2 + \frac{1}{x^2} \) and \( \rm x^3 + \frac{1}{x^3} \). The product of these two expressions is key:

\( \left( \rm x^2 + \frac{1}{x^2} \right) \left( \rm x^3 + \frac{1}{x^3} \right) = \rm x^2 \cdot x^3 + x^2 \cdot \frac{1}{x^3} + \frac{1}{x^2} \cdot x^3 + \frac{1}{x^2} \cdot \frac{1}{x^3} \)

\( = \rm x^5 + \frac{x^2}{x^3} + \frac{x^3}{x^2} + \frac{1}{x^5} \)

\( = \rm x^5 + \frac{1}{x} + x + \frac{1}{x^5} \)

\( = \left( \rm x^5 + \frac{1}{x^5} \right) + \left( \rm x + \frac{1}{x} \right) \)

So, we can rearrange this to find \( \rm x^5 + \frac{1}{x^5} \):

\( \rm x^5 + \frac{1}{x^5} = \left( x^2 + \frac{1}{x^2} \right) \left( x^3 + \frac{1}{x^3} \right) - \left( x + \frac{1}{x} \right) \)

Now, let's calculate the required terms:

Finding the Value of x² + 1/x²

Given \( \rm x + \frac{1}{x} = 4 \). Square both sides:

\( \left( \rm x + \frac{1}{x} \right)^2 = 4^2 \)

Using the identity \( (a+b)^2 = a^2 + b^2 + 2ab \):

\( \rm x^2 + \left( \frac{1}{x} \right)^2 + 2 \cdot x \cdot \frac{1}{x} = 16 \)

\( \rm x^2 + \frac{1}{x^2} + 2 = 16 \)

Subtract 2 from both sides:

\( \rm x^2 + \frac{1}{x^2} = 16 - 2 \)

\( \rm x^2 + \frac{1}{x^2} = 14 \)

Finding the Value of x³ + 1/x³

Given \( \rm x + \frac{1}{x} = 4 \). Cube both sides:

\( \left( \rm x + \frac{1}{x} \right)^3 = 4^3 \)

Using the identity \( (a+b)^3 = a^3 + b^3 + 3ab(a+b) \):

\( \rm x^3 + \left( \frac{1}{x} \right)^3 + 3 \cdot x \cdot \frac{1}{x} \left( x + \frac{1}{x} \right) = 64 \)

\( \rm x^3 + \frac{1}{x^3} + 3 \left( x + \frac{1}{x} \right) = 64 \)

Substitute the value of \( \rm x + \frac{1}{x} = 4 \):

\( \rm x^3 + \frac{1}{x^3} + 3(4) = 64 \)

\( \rm x^3 + \frac{1}{x^3} + 12 = 64 \)

Subtract 12 from both sides:

\( \rm x^3 + \frac{1}{x^3} = 64 - 12 \)

\( \rm x^3 + \frac{1}{x^3} = 52 \)

Calculating the Final Value of x⁵ + 1/x⁵

Now we use the relationship we derived:

\( \rm x^5 + \frac{1}{x^5} = \left( x^2 + \frac{1}{x^2} \right) \left( x^3 + \frac{1}{x^3} \right) - \left( x + \frac{1}{x} \right) \)

Substitute the values we found:

• \( \rm x^2 + \frac{1}{x^2} = 14 \)
• \( \rm x^3 + \frac{1}{x^3} = 52 \)
• \( \rm x + \frac{1}{x} = 4 \)

\( \rm x^5 + \frac{1}{x^5} = (14)(52) - 4 \)

First, calculate the product \( 14 \times 52 \):

\( \rm x^5 + \frac{1}{x^5} = 728 - 4 \)

\( \rm x^5 + \frac{1}{x^5} = 724 \)

Thus, the value of \( \rm x^5 + \frac{1}{x^5} \) is 724.

Revision Table: Key Identities and Values

Here's a quick summary of the values calculated and the identities used:

Additional Information: Generalizing xⁿ + 1/xⁿ

For problems involving finding \( \rm x^n + \frac{1}{x^n} \) when \( \rm x + \frac{1}{x} = k \), we can use a recursive formula or product methods:

1. For \( \rm x^2 + \frac{1}{x^2} \): \( \rm x^2 + \frac{1}{x^2} = k^2 - 2 \)
2. For \( \rm x^3 + \frac{1}{x^3} \): \( \rm x^3 + \frac{1}{x^3} = k^3 - 3k \)
3. For \( \rm x^4 + \frac{1}{x^4} \): \( \rm x^4 + \frac{1}{x^4} = (x^2 + \frac{1}{x^2})^2 - 2 = (k^2-2)^2 - 2 \)
4. For \( \rm x^5 + \frac{1}{x^5} \): \( \rm x^5 + \frac{1}{x^5} = (x^2 + \frac{1}{x^2})(x^3 + \frac{1}{x^3}) - (x + \frac{1}{x}) = (k^2-2)(k^3-3k) - k \)

Knowing these patterns can help solve similar problems quickly. The general method involves finding ways to express \( x^n + \frac{1}{x^n} \) in terms of lower powers like \( x + \frac{1}{x} \), \( x^2 + \frac{1}{x^2} \), etc.