If `(13^(2) - 5^(2))^(3//2) = 6^(3) xx A`, then the value of A is

Options

A. 2
B. `2^(4)`
C. `2^(2)`
D. `2^(3)`

Answer

To solve the equation \((13^2 - 5^2)^{3/2} = 6^3 \times A\), we will follow these steps:

### Step 1: Calculate \(13^2\) and \(5^2\)
First, we need to calculate the squares of 13 and 5:
\[ 13^2 = 169 \]
\[ 5^2 = 25 \]

### Step 2: Subtract the squares
Now, we subtract \(5^2\) from \(13^2\):
\[ 13^2 - 5^2 = 169 - 25 = 144 \]

### Step 3: Raise the result to the power of \(3/2\)
Next, we raise the result to the power of \(3/2\):
\[ (144)^{3/2} \]

### Step 4: Simplify \(144^{3/2}\)
We can simplify \(144^{3/2}\) as follows:
\[ 144^{3/2} = (144^{1/2})^3 = (12)^3 = 1728 \]

### Step 5: Set up the equation
Now we can set up the equation:
\[ 1728 = 6^3 \times A \]

### Step 6: Calculate \(6^3\)
Next, we calculate \(6^3\):
\[ 6^3 = 216 \]

### Step 7: Solve for \(A\)
Now we can solve for \(A\):
\[ A = \frac{1728}{216} \]

### Step 8: Simplify \(\frac{1728}{216}\)
Now we simplify:
\[ A = 8 \]

### Step 9: Final answer
Thus, the value of \(A\) is:
\[ A = 8 \]