Certainly! Let’s solve the expression [tex]\(\frac{a^b}{a^{a-b}}\)[/tex] step-by-step.
First, recall the properties of exponents. One of the key properties we will use is:
[tex]\[ \frac{x^m}{x^n} = x^{m-n} \][/tex]
Given the expression:
[tex]\[ \frac{a^b}{a^{a-b}} \][/tex]
We can rewrite it using the property of exponents mentioned above:
[tex]\[ \frac{a^b}{a^{a-b}} = a^{b - (a-b)} \][/tex]
Now, let's simplify the exponent:
[tex]\[ b - (a - b) \][/tex]
We can distribute the negative sign inside the parentheses:
[tex]\[ b - a + b \][/tex]
Combine the like terms:
[tex]\[ b + b - a = 2b - a \][/tex]
So, the simplified form of the original expression [tex]\(\frac{a^b}{a^{a-b}}\)[/tex] is:
[tex]\[ a^{2b - a} \][/tex]
Therefore, the final answer is:
[tex]\[ a^{2b - a} \][/tex]
