Sure, let's simplify the expression [tex]\(\frac{a^b}{a^{a-b}}\)[/tex] step-by-step.
1. Rewrite the expression using the properties of exponents:
The given expression is [tex]\(\frac{a^b}{a^{a-b}}\)[/tex]. According to the properties of exponents, specifically the quotient rule which states that [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex], we can rewrite the expression as:
[tex]\[
\frac{a^b}{a^{a-b}} = a^{b - (a - b)}
\][/tex]
2. Simplify the exponent:
In the exponent [tex]\(b - (a - b)\)[/tex], distribute the negative sign inside the parentheses:
[tex]\[
b - (a - b) = b - a + b
\][/tex]
Combine like terms:
[tex]\[
b - a + b = 2b - a
\][/tex]
3. Write the simplified expression:
Substitute the simplified exponent back into the expression:
[tex]\[
a^{b - (a - b)} = a^{2b - a}
\][/tex]
Thus, the simplified form of the expression [tex]\(\frac{a^b}{a^{a-b}}\)[/tex] is:
[tex]\[
a^{2b - a}
\][/tex]
So, [tex]\(\frac{a^b}{a^{a-b}} = a^{2b - a}\)[/tex].
