If $\frac{x^{a}}{x^{b}}=x^{8}$ and $a+b=7$, what is the value of $a^{2}-b^{2} ?$ A) 1 B) $\frac{8}{7}$ C) 15 D) 56

Step 1 ：We are given the equation \(\frac{x^{a}}{x^{b}}=x^{8}\) and the condition that \(a+b=7\).

Step 2 ：Using the properties of exponents, we can rewrite the given equation as \(x^{a-b}=x^{8}\).

Step 3 ：Since the bases are the same, the exponents must be equal. Therefore, we have \(a-b=8\).

Step 4 ：Now we have two equations, \(a-b=8\) and \(a+b=7\).

Step 5 ：Solving these two equations, we find that \(a=\frac{15}{2}\) and \(b=-\frac{1}{2}\).

Step 6 ：We are asked to find the value of \(a^{2}-b^{2}\). Substituting the values of \(a\) and \(b\) into this expression, we get 56.

Step 7 ：Final Answer: The value of \(a^{2}-b^{2}\) is \(\boxed{56}\).