If $x+\frac{1}{x}$=5, then what is the value of $x^3+\frac{1}{x^3}$ ?

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

To solve for the value of $ x^3 + \frac{1}{x^3} $ given that $ x + \frac{1}{x} = 5 $, we can use algebraic identities and manipulation. ### Step-by-step Explanation: 1. **Given Equation:** \[ x + \frac{1}{x} = 5 \] 2. **Square Both Sides:** \[ \left(x + \frac{1}{x}\right)^2 = 5^2 \] \[ x^2 + 2 + \frac{1}{x^2} = 25 \] \[ x^2 + \frac{1}{x^2} = 25 - 2 = 23 \] 3. **Cube Both Sides of the Original Equation:** \[ \left(x + \frac{1}{x}\right)^3 = 5^3 \] \[ x^3 + 3x + \frac{3}{x} + \frac{1}{x^3} = 125 \] 4. **Substitute $ x + \frac{1}{x} = 5 $ into the Cubed Equation:** \[ x^3 + \frac{1}{x^3} + 3\left(x + \frac{1}{x}\right) = 125 \] \[ x^3 + \frac{1}{x^3} + 3 \times 5 = 125 \] \[ x^3 + \frac{1}{x^3} + 15 = 125 \] 5. **Solve for $ x^3 + \frac{1}{x^3} $:** \[ x^3 + \frac{1}{x^3} = 125 - 15 = 110 \] ### Conclusion: - The value of $ x^3 + \frac{1}{x^3} $ is indeed 110, which corresponds to **Option 2**. ### Explanation of Incorrect Options: - **Option 1 (125):** This is the value of $ \left(x + \frac{1}{x}\right)^3 $, not $ x^3 + \frac{1}{x^3} $. - **Option 3 (115):** This value does not satisfy the derived equation from the given conditions. - **Option 4 (105):** This value is also incorrect as it does not match the calculated result. Thus, the correct answer is **Option 2 (110)**, as derived through logical and mathematical reasoning.

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 and manipulation.

### Step-by-step Explanation:

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

2. **Square Both Sides:**
   $$
   \left(x + \frac{1}{x}\right)^2 = 5^2
   $$
   $$
   x^2 + 2 + \frac{1}{x^2} = 25
   $$
   $$
   x^2 + \frac{1}{x^2} = 25 - 2 = 23
   $$

3. **Cube Both Sides of the Original Equation:**
   $$
   \left(x + \frac{1}{x}\right)^3 = 5^3
   $$
   $$
   x^3 + 3x + \frac{3}{x} + \frac{1}{x^3} = 125
   $$

4. **Substitute $ x + \frac{1}{x} = 5 $ into the Cubed Equation:**
   $$
   x^3 + \frac{1}{x^3} + 3\left(x + \frac{1}{x}\right) = 125
   $$
   $$
   x^3 + \frac{1}{x^3} + 3 \times 5 = 125
   $$
   $$
   x^3 + \frac{1}{x^3} + 15 = 125
   $$

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

### Conclusion:

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

### Explanation of Incorrect Options:

- **Option 1 (125):** This is the value of $ \left(x + \frac{1}{x}\right)^3 $, not $ x^3 + \frac{1}{x^3} $.
- **Option 3 (115):** This value does not satisfy the derived equation from the given conditions.
- **Option 4 (105):** This value is also incorrect as it does not match the calculated result.

Thus, the correct answer is **Option 2 (110)**, as derived through logical and mathematical reasoning.