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 \((13^2 - 5^2)^{\frac{3}{2}} = 6^3 \times A\), 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 \(5^2\) from \(13^2\)
Now, we subtract \(5^2\) from \(13^2\):
\[ 13^2 - 5^2 = 169 - 25 = 144 \]

### Step 3: Raise the result to the power of \(\frac{3}{2}\)
Next, we raise the result to the power of \(\frac{3}{2}\):
\[ (144)^{\frac{3}{2}} = (12^2)^{\frac{3}{2}} = 12^{2 \times \frac{3}{2}} = 12^3 \]

### Step 4: Calculate \(12^3\)
Now, we calculate \(12^3\):
\[ 12^3 = 1728 \]

### Step 5: Set up the equation
Now we can set up the equation:
\[ 12^3 = 6^3 \times A \]
We know that \(6^3 = 216\), so we can rewrite the equation:
\[ 1728 = 216 \times A \]

### Step 6: Solve for \(A\)
To find \(A\), we divide both sides by 216:
\[ A = \frac{1728}{216} \]

### Step 7: Simplify the fraction
Now we simplify the fraction:
\[ A = 8 \]

Thus, the value of \(A\) is \(8\).

### Summary of Steps:
1. Calculate \(13^2\) and \(5^2\).
2. Subtract \(5^2\) from \(13^2\).
3. Raise the result to the power of \(\frac{3}{2}\).
4. Calculate \(12^3\).
5. Set up the equation with \(6^3\).
6. Solve for \(A\) by dividing.
7. Simplify to find the final value of \(A\).