If `int (dx)/( sqrt( 8 ax - x ^(2))) = a ^(m) sin ^(-1) ((x - 4a)/( 4a)) ` 'x' and 'a' represent distance, then find the value of 'm' using dimensional analysis.

Explanation

To solve the given problem using dimensional analysis, we will analyze both sides of the equation step by step.

### Step 1: Identify the Left-Hand Side (LHS)
The LHS of the equation is given as:
\[ \int \frac{dx}{\sqrt{8ax - x^2}} \]

- Here, \(dx\) represents a differential element of distance, which has the dimension of length \([L]\).
- The expression under the square root is \(8ax - x^2\). Since both \(a\) and \(x\) represent distances, their dimensions are also \([L]\).

Now, let's analyze the term inside the square root:
\[ 8ax - x^2 \]
- The term \(8ax\) has dimensions \([L][L] = [L^2]\).
- The term \(x^2\) also has dimensions \([L^2]\).

Thus, the expression \(8ax - x^2\) has the dimension of \([L^2]\).

Now, taking the square root:
\[ \sqrt{8ax - x^2} \text{ has dimensions } [L] \]

Now, substituting this back into the LHS:
\[ \int \frac{dx}{\sqrt{8ax - x^2}} \text{ has dimensions } \frac{[L]}{[L]} = [1] \]

### Step 2: Identify the Right-Hand Side (RHS)
The RHS is given as:
\[ a^m \sin^{-1}\left(\frac{x - 4a}{4a}\right) \]

- The term \(a^m\) has dimensions \([L]^m\).
- The term \(\sin^{-1}\) is a dimensionless quantity (it is a ratio), so it has dimensions \([1]\).

Thus, the dimensions of the RHS can be expressed as:
\[ [L]^m \cdot [1] = [L]^m \]

### Step 3: Set the Dimensions Equal
Since the LHS and RHS must be equal, we can set their dimensions equal to each other:
\[ [1] = [L]^m \]

### Step 4: Solve for \(m\)
To satisfy the equation, the only way for \([L]^m\) to equal \([1]\) is if \(m = 0\):
\[ m = 0 \]

### Conclusion
Thus, the value of \(m\) is:
\[ \boxed{0} \]