Understanding the Problem: Finding \( \rm x^5+\frac{1}{x^5} \)

The question asks us to find the value of the expression \( \rm x^5+\frac{1}{x^5} \) given that \( \rm x+\frac{1}{x}=-6 \). This type of problem often involves using algebraic identities to express higher powers in terms of lower powers.

Step-by-Step Calculation of \( \rm x^5+\frac{1}{x^5} \)

We are given the value of \( \rm x+\frac{1}{x} \). To find \( \rm x^5+\frac{1}{x^5} \), we can find the values of \( \rm x^2+\frac{1}{x^2} \) and \( \rm x^3+\frac{1}{x^3} \) first. The product of these two expressions will help us find the required value.

Finding the Value of \( \rm x^2+\frac{1}{x^2} \)

We start with the given equation:

\( \rm x+\frac{1}{x}=-6 \)

Square both sides of the equation:

\( \rm \left(x+\frac{1}{x}\right)^2 = (-6)^2 \)

Using the identity \( \rm (a+b)^2 = a^2+2ab+b^2 \), we get:

\( \rm x^2 + 2 \cdot x \cdot \frac{1}{x} + \left(\frac{1}{x}\right)^2 = 36 \)

\( \rm x^2 + 2 + \frac{1}{x^2} = 36 \)

Subtract 2 from both sides:

\( \rm x^2 + \frac{1}{x^2} = 36 - 2 \)

\( \rm x^2 + \frac{1}{x^2} = 34 \)

Finding the Value of \( \rm x^3+\frac{1}{x^3} \)

We use the original equation again:

\( \rm x+\frac{1}{x}=-6 \)

Cube both sides of the equation:

\( \rm \left(x+\frac{1}{x}\right)^3 = (-6)^3 \)

Using the identity \( \rm (a+b)^3 = a^3+b^3+3ab(a+b) \), we get:

\( \rm x^3 + \left(\frac{1}{x}\right)^3 + 3 \cdot x \cdot \frac{1}{x} \left(x+\frac{1}{x}\right) = -216 \)

\( \rm x^3 + \frac{1}{x^3} + 3 \left(x+\frac{1}{x}\right) = -216 \)

Substitute the given value \( \rm x+\frac{1}{x} = -6 \):

\( \rm x^3 + \frac{1}{x^3} + 3(-6) = -216 \)

\( \rm x^3 + \frac{1}{x^3} - 18 = -216 \)

Add 18 to both sides:

\( \rm x^3 + \frac{1}{x^3} = -216 + 18 \)

\( \rm x^3 + \frac{1}{x^3} = -198 \)

Finding the Value of \( \rm x^5+\frac{1}{x^5} \)

Now we have the values for \( \rm x^2+\frac{1}{x^2} \) and \( \rm x^3+\frac{1}{x^3} \). Consider the product of these two expressions:

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

Let's expand this product:

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

\( \rm = x^{2+3} + x^{2-3} + x^{3-2} + x^{-2-3} \)

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

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

Rearranging the terms, we get:

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

So, we have the relationship:

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

We know the values of \( \rm x^2+\frac{1}{x^2} \) (which is 34), \( \rm x^3+\frac{1}{x^3} \) (which is -198), and \( \rm x+\frac{1}{x} \) (which is -6). Substitute these values into the equation:

\( \rm (34)(-198) = x^5 + \frac{1}{x^5} + (-6) \)

Calculate the product \( \rm 34 \times -198 \):

\( \rm 34 \times 198 = 6732 \)

So, \( \rm 34 \times -198 = -6732 \).

The equation becomes:

\( \rm -6732 = x^5 + \frac{1}{x^5} - 6 \)

To find \( \rm x^5 + \frac{1}{x^5} \), add 6 to both sides of the equation:

\( \rm x^5 + \frac{1}{x^5} = -6732 + 6 \)

\( \rm x^5 + \frac{1}{x^5} = -6726 \)

Summary of Calculations

The calculated value for \( \rm x^5+\frac{1}{x^5} \) is -6726.

Revision Table: Key Algebraic Identities

These identities are fundamental when dealing with expressions involving sums and powers of variables, especially in the form of \( \rm x+\frac{1}{x} \).

Additional Information on \( \rm x^n+\frac{1}{x^n} \) Problems

Problems involving \( \rm x+\frac{1}{x} \) and finding \( \rm x^n+\frac{1}{x^n} \) for integer values of n are common in algebra. The general approach involves using lower power results to find higher power results.

• To find \( \rm x^2+\frac{1}{x^2} \), square \( \rm x+\frac{1}{x} \).
• To find \( \rm x^3+\frac{1}{x^3} \), cube \( \rm x+\frac{1}{x} \).
• To find \( \rm x^4+\frac{1}{x^4} \), square \( \rm x^2+\frac{1}{x^2} \).
• To find \( \rm x^5+\frac{1}{x^5} \), use the product \( \rm (x^2+\frac{1}{x^2})(x^3+\frac{1}{x^3}) \).
• To find \( \rm x^6+\frac{1}{x^6} \), either cube \( \rm x^2+\frac{1}{x^2} \) or square \( \rm x^3+\frac{1}{x^3} \).

Understanding these relationships and the core algebraic identities allows you to solve these problems efficiently.