If `a ** b ** c = sqrt(((a+2)(b +3))/((c+1)))`, then the value of `(6 ** 15 ** 3)` is :

Options

A. 6
B. 3
C. 4
D. can't be determined

Answer

To solve the equation \( a^{b^{c}} = \sqrt{\frac{(a+2)(b+3)}{(c+1)}} \) for the values \( a = 6 \), \( b = 15 \), and \( c = 3 \), we will follow these steps:

### Step 1: Substitute the values into the equation
We start by substituting \( a = 6 \), \( b = 15 \), and \( c = 3 \) into the equation:

\[ 6^{15^{3}} = \sqrt{\frac{(6+2)(15+3)}{(3+1)}} \]

### Step 2: Simplify the expression inside the square root
Now, we simplify the expression inside the square root:

\[ (6 + 2) = 8, \quad (15 + 3) = 18, \quad (3 + 1) = 4 \]

So, we can rewrite the equation as:

\[ 6^{15^{3}} = \sqrt{\frac{8 \cdot 18}{4}} \]

### Step 3: Calculate the numerator and denominator
Next, we calculate the numerator and denominator:

\[ 8 \cdot 18 = 144 \]

Now, we divide by the denominator:

\[ \frac{144}{4} = 36 \]

### Step 4: Take the square root
Now we take the square root of 36:

\[ \sqrt{36} = 6 \]

### Step 5: Compare both sides of the equation
Now we compare both sides of the equation:

\[ 6^{15^{3}} = 6 \]

Since \( 6^{15^{3}} \) is not equal to 6, we realize that the left side is much larger than the right side. However, the question seems to imply that we are looking for the value of the right side, which is \( 6 \).

### Final Answer
Thus, the value of \( (6^{15^{3}}) \) is:

\[ \boxed{6} \]