Find the value of : $\sqrt[4]{81} + \sqrt[3]{216} + \sqrt[5]{32}$

Q: Find the value of : $\sqrt[4]{81} + \sqrt[3]{216} + \sqrt[5]{32}$

To find the value of $\sqrt[4]{81} + \sqrt[3]{216} + \sqrt[5]{32}$, we need to evaluate each term separately and then sum them up. 1. **Evaluate $\sqrt[4]{81}$:** \[ \sqrt[4]{81} = \sqrt[4]{3^4} = 3 \] Since $81 = 3^4$, the fourth root of 81 is 3. 2. **Evaluate $\sqrt[3]{216}$:** \[ \sqrt[3]{216} = \sqrt[3]{6^3} = 6 \] Since $216 = 6^3$, the cube root of 216 is 6. 3. **Evaluate $\sqrt[5]{32}$:** \[ \sqrt[5]{32} = \sqrt[5]{2^5} = 2 \] Since $32 = 2^5$, the fifth root of 32 is 2. Now, add the results of these evaluations: \[ 3 + 6 + 2 = 11 \] Thus, the value of $\sqrt[4]{81} + \sqrt[3]{216} + \sqrt[5]{32}$ is 11. **Explanation of Options:** - **Option 1 (10):** This is incorrect because the sum of the evaluated terms is 11, not 10. - **Option 2 (11):** This is the correct answer, as shown by the calculations above. - **Option 3 (12):** This is incorrect because the sum of the evaluated terms is 11, not 12. - **Option 4 (32):** This is incorrect because the sum of the evaluated terms is 11, not 32. Therefore, the correct answer is Option 2, which is 11.

Question

Accepted Answer

To find the value of $\sqrt[4]{81} + \sqrt[3]{216} + \sqrt[5]{32}$, we need to evaluate each term separately and then sum them up.

1. **Evaluate $\sqrt[4]{81}$:**

$$
   \sqrt[4]{81} = \sqrt[4]{3^4} = 3
   $$

Since $81 = 3^4$, the fourth root of 81 is 3.

2. **Evaluate $\sqrt[3]{216}$:**

$$
   \sqrt[3]{216} = \sqrt[3]{6^3} = 6
   $$

Since $216 = 6^3$, the cube root of 216 is 6.

3. **Evaluate $\sqrt[5]{32}$:**

$$
   \sqrt[5]{32} = \sqrt[5]{2^5} = 2
   $$

Since $32 = 2^5$, the fifth root of 32 is 2.

Now, add the results of these evaluations:

$$
3 + 6 + 2 = 11
$$

Thus, the value of $\sqrt[4]{81} + \sqrt[3]{216} + \sqrt[5]{32}$ is 11.

**Explanation of Options:**

- **Option 1 (10):** This is incorrect because the sum of the evaluated terms is 11, not 10.
- **Option 2 (11):** This is the correct answer, as shown by the calculations above.
- **Option 3 (12):** This is incorrect because the sum of the evaluated terms is 11, not 12.
- **Option 4 (32):** This is incorrect because the sum of the evaluated terms is 11, not 32.

Therefore, the correct answer is Option 2, which is 11.