If $a - \frac{1}{a} = \sqrt{21}$, where a 0, then the value of $\left( a^2 + \frac{1}{a^2} \right) \left( a + \frac{1}{a} \right)$ is equal to:

Question

If $a - \frac{1}{a} = \sqrt{21}$, where a > 0, then the value of $\left( a^2 + \frac{1}{a^2} \right) \left( a + \frac{1}{a} \right)$ is equal to:

Accepted Answer

To solve the problem and justify why Option 2 is the correct answer, we need to find the value of $\left( a^2 + \frac{1}{a^2} \right) \left( a + \frac{1}{a} \right)$ given that $a - \frac{1}{a} = \sqrt{21}$.

### Step-by-step Explanation:

1. **Given Equation:**
   $$
   a - \frac{1}{a} = \sqrt{21}
   $$

2. **Find $a + \frac{1}{a}$:**
   We know that:
   $$
   (a + \frac{1}{a})^2 = a^2 + 2 + \frac{1}{a^2}
   $$
   and
   $$
   (a - \frac{1}{a})^2 = a^2 - 2 + \frac{1}{a^2}
   $$

Given $a - \frac{1}{a} = \sqrt{21}$, we have:
   $$
   (a - \frac{1}{a})^2 = 21
   $$
   $$
   a^2 - 2 + \frac{1}{a^2} = 21
   $$

Therefore:
   $$
   a^2 + \frac{1}{a^2} = 23
   $$

3. **Calculate $(a^2 + \frac{1}{a^2})(a + \frac{1}{a})$:**
   We need to find $a + \frac{1}{a}$. Using the identity:
   $$
   (a + \frac{1}{a})^2 = a^2 + 2 + \frac{1}{a^2}
   $$
   $$
   a^2 + \frac{1}{a^2} = 23 \implies (a + \frac{1}{a})^2 = 25
   $$
   $$
   a + \frac{1}{a} = \sqrt{25} = 5
   $$

4. **Final Calculation:**
   $$
   (a^2 + \frac{1}{a^2})(a + \frac{1}{a}) = 23 \times 5 = 115
   $$

Thus, the value of $\left( a^2 + \frac{1}{a^2} \right) \left( a + \frac{1}{a} \right)$ is indeed 115, which corresponds to Option 2.

### Explanation for Incorrect Options:

- **Option 1 (113):** This value does not match the calculated result of 115. The calculations clearly show that the correct product is 115, not 113.

- **Option 3 (114):** Similarly, 114 is not the result of the calculations. The step-by-step process confirms that the correct value is 115.

- **Option 4 (110):** This option is also incorrect as it does not match the derived value of 115 from the given conditions.

Therefore, based on the calculations and logical reasoning, Option 2 (115) is the correct answer.