The value of $\sqrt{16}+\sqrt[3]{64}+\sqrt[4]{256}$ is:-

Question

Accepted Answer

The correct option is 4:
To find the value of the expression $\sqrt{16}+\sqrt$$3$${64}+\sqrt$$4$${256}$, we need to evaluate each radical term individually.

**Key Concepts:**  * **Square Root:** The square root of a number $x$ is a number $y$ such that $y^2 = x$. We are looking for the principal (positive) square root.
  * **Cube Root:** The cube root of a number $x$ is a number $y$ such that $y^3 = x$.
  * **Fourth Root:** The fourth root of a number $x$ is a number $y$ such that $y^4 = x$. We are looking for the principal (positive) fourth root.**Step-by-step Explanation:**1.  **Evaluate $\sqrt{16}$:**
    We need a number that, when multiplied by itself, equals 16.
    Since $4 \times 4 = 16$, we have $\sqrt{16} = 4$.

2.  **Evaluate $\sqrt$$3$${64}$:**
    We need a number that, when multiplied by itself three times, equals 64.
    Since $4 \times 4 \times 4 = 16 \times 4 = 64$, we have $\sqrt$$3$${64} = 4$.

3.  **Evaluate $\sqrt$$4$${256}$:**
    We need a number that, when multiplied by itself four times, equals 256.
    We know that $4^2 = 16$, and $4^3 = 64$.
    Then, $4^4 = 4^3 \times 4 = 64 \times 4 = 256$.
    So, $\sqrt$$4$${256} = 4$.

4.  **Calculate the sum:**
    Now, substitute the values back into the original expression:
    $\sqrt{16}+\sqrt$$3$${64}+\sqrt$$4$${256} = 4 + 4 + 4 = 12$.Therefore, the value of the expression is 12.

The incorrect option is 1 (24): This option would be obtained if there was a calculation error, such as mistaking the values of the roots or incorrectly adding them. For example, if one mistakenly thought $\sqrt[3]{64}$ or $\sqrt[4]{256}$ was a much larger number.

The incorrect option is 2 (28): This option also results from an arithmetic error in calculating the roots or their sum. There is no clear mathematical operation that leads to 28 from the given expression.

The incorrect option is 3 (18): This option would result from an error in calculating one or more of the roots or in the final summation. For example, if one of the roots was incorrectly evaluated.