The value of $\sqrt[3]{1 + \sqrt[3]{343}}$ is:

Q: The value of $\sqrt[3]{1 + \sqrt[3]{343}}$ is:

To determine the value of $\sqrt[3]{1 + \sqrt[3]{343}}$, we need to evaluate the expression step by step. 1. **Evaluate $\sqrt[3]{343}$:** The cube root of 343 is calculated as follows: \[ 343 = 7^3 \] Therefore, $\sqrt[3]{343} = 7$. 2. **Substitute back into the expression:** Now substitute $\sqrt[3]{343} = 7$ back into the original expression: \[ \sqrt[3]{1 + \sqrt[3]{343}} = \sqrt[3]{1 + 7} \] 3. **Simplify the expression:** Simplify inside the cube root: \[ 1 + 7 = 8 \] 4. **Evaluate $\sqrt[3]{8}$:** The cube root of 8 is calculated as follows: \[ 8 = 2^3 \] Therefore, $\sqrt[3]{8} = 2$. Thus, the value of $\sqrt[3]{1 + \sqrt[3]{343}}$ is 2, which corresponds to Option 2. **Explanation of why other options are incorrect:** - **Option 1 (1):** This option is incorrect because the cube root of 8 is not 1. The cube root of 8 is 2, as shown in the calculations above. - **Option 3 (3):** This option is incorrect because the cube root of 8 is not 3. The cube root of 8 is 2, as demonstrated in the calculations. - **Option 4 (7):** This option is incorrect because it seems to confuse the cube root of 343 with the final result. While $\sqrt[3]{343} = 7$, the expression $\sqrt[3]{1 + \sqrt[3]{343}}$ simplifies to $\sqrt[3]{8}$, which is 2, not 7. Therefore, the correct answer is indeed Option 2, as the calculations and logical reasoning confirm that $\sqrt[3]{1 + \sqrt[3]{343}} = 2$.

Question

Accepted Answer

To determine the value of $\sqrt[3]{1 + \sqrt[3]{343}}$, we need to evaluate the expression step by step.

1. **Evaluate $\sqrt[3]{343}$:**

The cube root of 343 is calculated as follows:
   $$
   343 = 7^3
   $$
   Therefore, $\sqrt[3]{343} = 7$.

2. **Substitute back into the expression:**

Now substitute $\sqrt[3]{343} = 7$ back into the original expression:
   $$
   \sqrt[3]{1 + \sqrt[3]{343}} = \sqrt[3]{1 + 7}
   $$

3. **Simplify the expression:**

Simplify inside the cube root:
   $$
   1 + 7 = 8
   $$

4. **Evaluate $\sqrt[3]{8}$:**

The cube root of 8 is calculated as follows:
   $$
   8 = 2^3
   $$
   Therefore, $\sqrt[3]{8} = 2$.

Thus, the value of $\sqrt[3]{1 + \sqrt[3]{343}}$ is 2, which corresponds to Option 2.

**Explanation of why other options are incorrect:**

- **Option 1 (1):** This option is incorrect because the cube root of 8 is not 1. The cube root of 8 is 2, as shown in the calculations above.

- **Option 3 (3):** This option is incorrect because the cube root of 8 is not 3. The cube root of 8 is 2, as demonstrated in the calculations.

- **Option 4 (7):** This option is incorrect because it seems to confuse the cube root of 343 with the final result. While $\sqrt[3]{343} = 7$, the expression $\sqrt[3]{1 + \sqrt[3]{343}}$ simplifies to $\sqrt[3]{8}$, which is 2, not 7.

Therefore, the correct answer is indeed Option 2, as the calculations and logical reasoning confirm that $\sqrt[3]{1 + \sqrt[3]{343}} = 2$.