To solve the expression [tex]\(\frac{a^b}{a^{a-b}}\)[/tex], we'll use properties of exponents to simplify it step-by-step.
### Step 1: Use the properties of exponents
We know one of the key properties of exponents is:
[tex]\[
\frac{x^m}{x^n} = x^{m-n}
\][/tex]
### Step 2: Identify [tex]\(m\)[/tex] and [tex]\(n\)[/tex]
In our given expression, [tex]\(x = a\)[/tex], [tex]\(m = b\)[/tex], and [tex]\(n = a - b\)[/tex]. Therefore, our expression [tex]\(\frac{a^b}{a^{a-b}}\)[/tex] can be rewritten using the property of exponents from Step 1:
[tex]\[
\frac{a^b}{a^{a-b}} = a^{b - (a - b)}
\][/tex]
### Step 3: Simplify the exponent
Simplify the exponent [tex]\(b - (a - b)\)[/tex]:
[tex]\[
b - (a - b) = b - a + b = 2b - a
\][/tex]
### Step 4: Substitute back into the base
Now we have:
[tex]\[
a^{b - (a - b)} = a^{2b - a}
\][/tex]
So, the simplified form of the expression [tex]\(\frac{a^b}{a^{a-b}}\)[/tex] is:
[tex]\[
a^{2b - a}
\][/tex]
Hence, [tex]\(\frac{a^b}{a^{a-b}} = a^{2b - a}\)[/tex].
