If ($x + \frac{1}{x}$ = 5, then the value of ($x^3 + \frac{1}{x^3})$ is:

Q: If ($x + \frac{1}{x}$ = 5, then the value of ($x^3 + \frac{1}{x^3})$ is:

To solve for the value of $x^3 + \frac{1}{x^3}$ given that $x + \frac{1}{x} = 5$, we can use algebraic identities to find the solution. ### Explanation: 1. **Given:** \[ x + \frac{1}{x} = 5 \] 2. **Objective:** Find the value of $x^3 + \frac{1}{x^3}$. 3. **Using the identity:** \[ x^3 + \frac{1}{x^3} = \left(x + \frac{1}{x}\right)^3 - 3\left(x + \frac{1}{x}\right) \] 4. **Substitute the given value:** \[ \left(x + \frac{1}{x}\right)^3 = 5^3 = 125 \] 5. **Calculate:** \[ x^3 + \frac{1}{x^3} = 125 - 3 \times 5 \] \[ x^3 + \frac{1}{x^3} = 125 - 15 = 110 \] Thus, the value of $x^3 + \frac{1}{x^3}$ is 110, which corresponds to **Option 2**. ### Why Other Options Are Incorrect: - **Option 1 (125):** This is the value of $\left(x + \frac{1}{x}\right)^3$ before subtracting $3(x + \frac{1}{x})$. It does not account for the subtraction step in the identity used to find $x^3 + \frac{1}{x^3}$. - **Option 3 (25):** This value does not relate to any step in the calculation process for $x^3 + \frac{1}{x^3}$ given the initial condition. It is not derived from any logical step in the identity used. - **Option 4 (5):** This is the initial given value of $x + \frac{1}{x}$, not the result of the calculation for $x^3 + \frac{1}{x^3}$. Therefore, based on the algebraic manipulation and identity used, **Option 2 (110)** is the correct answer.

Question

Accepted Answer

To solve for the value of $x^3 + \frac{1}{x^3}$ given that $x + \frac{1}{x} = 5$, we can use algebraic identities to find the solution.

### Explanation:

1. **Given:**
   $$
   x + \frac{1}{x} = 5
   $$

2. **Objective:**
   Find the value of $x^3 + \frac{1}{x^3}$.

3. **Using the identity:**
   $$
   x^3 + \frac{1}{x^3} = \left(x + \frac{1}{x}\right)^3 - 3\left(x + \frac{1}{x}\right)
   $$

4. **Substitute the given value:**
   $$
   \left(x + \frac{1}{x}\right)^3 = 5^3 = 125
   $$

5. **Calculate:**
   $$
   x^3 + \frac{1}{x^3} = 125 - 3 \times 5
   $$
   $$
   x^3 + \frac{1}{x^3} = 125 - 15 = 110
   $$

Thus, the value of $x^3 + \frac{1}{x^3}$ is 110, which corresponds to **Option 2**.

### Why Other Options Are Incorrect:

- **Option 1 (125):** This is the value of $\left(x + \frac{1}{x}\right)^3$ before subtracting $3(x + \frac{1}{x})$. It does not account for the subtraction step in the identity used to find $x^3 + \frac{1}{x^3}$.

- **Option 3 (25):** This value does not relate to any step in the calculation process for $x^3 + \frac{1}{x^3}$ given the initial condition. It is not derived from any logical step in the identity used.

- **Option 4 (5):** This is the initial given value of $x + \frac{1}{x}$, not the result of the calculation for $x^3 + \frac{1}{x^3}$.

Therefore, based on the algebraic manipulation and identity used, **Option 2 (110)** is the correct answer.