If `2 ** 3 =sqrt(13)` and `3 ** 4=5`, then the value of `5** 12` is

Options

A. 17
B. `sqrt(29)`
C. 21
D. 13

Answer

To solve the problem step by step, we will first understand the operation defined by the "star" operation and then apply it to find the value of \(5 \star 12\).

### Step 1: Understand the "star" operation
The operation \(a \star b\) is defined as:
\[ a \star b = \sqrt{a^2 + b^2} \]
This means that for any two numbers \(a\) and \(b\), we take the square of each number, add them together, and then take the square root of the sum.

### Step 2: Apply the operation to \(5 \star 12\)
Now, we need to find \(5 \star 12\):
\[ 5 \star 12 = \sqrt{5^2 + 12^2} \]

### Step 3: Calculate \(5^2\) and \(12^2\)
First, we calculate the squares:
\[ 5^2 = 25 \]
\[ 12^2 = 144 \]

### Step 4: Add the squares
Next, we add the results:
\[ 5^2 + 12^2 = 25 + 144 = 169 \]

### Step 5: Take the square root
Now, we take the square root of the sum:
\[ 5 \star 12 = \sqrt{169} \]

### Step 6: Simplify the square root
Finally, we simplify the square root:
\[ \sqrt{169} = 13 \]

### Conclusion
Thus, the value of \(5 \star 12\) is:
\[ \boxed{13} \]