Certainly! Let's simplify the given expression step-by-step:
The expression we need to simplify is:
[tex]\[
\frac{a^{n-m} \cdot a^{2m-2n}}{a^{m-n}}
\][/tex]
### Step 1: Combine the exponents in the numerator
Using the property of exponents [tex]\( a^x \cdot a^y = a^{x+y} \)[/tex], we can combine the exponents in the numerator:
[tex]\[
a^{n-m} \cdot a^{2m-2n} = a^{(n-m) + (2m-2n)}
\][/tex]
### Step 2: Simplify the exponent in the numerator
Now, add the exponents in the numerator:
[tex]\[
(n - m) + (2m - 2n) = n - m + 2m - 2n
\][/tex]
Combine the terms to simplify further:
[tex]\[
n - m + 2m - 2n = (n - 2n) + (2m - m) = -n + m
\][/tex]
So, the numerator simplifies to:
[tex]\[
a^{-n+m}
\][/tex]
### Step 3: Simplify the entire expression
Now we have:
[tex]\[
\frac{a^{-n+m}}{a^{m-n}}
\][/tex]
Using the property of exponents [tex]\( \frac{a^x}{a^y} = a^{x-y} \)[/tex], we can combine the exponents in the entire expression:
[tex]\[
a^{-n+m} \div a^{m-n} = a^{(-n+m) - (m-n)}
\][/tex]
### Step 4: Simplify the exponent in the entire expression
Now, simplify the exponent:
[tex]\[
(-n + m) - (m - n) = -n + m - m + n
\][/tex]
Combine the terms:
[tex]\[
-n + m - m + n = (-n + n) + (m - m) = 0
\][/tex]
So, the expression simplifies to:
[tex]\[
a^0
\][/tex]
### Step 5: Evaluate [tex]\( a^0 \)[/tex]
By definition, any non-zero number raised to the power of 0 is 1:
[tex]\[
a^0 = 1
\][/tex]
Thus, the simplified expression is:
[tex]\[
\boxed{1}
\][/tex]
So, the final simplified form of the given expression is [tex]\( \boxed{1} \)[/tex].
