Sure, let’s simplify the given expression step-by-step:
[tex]\[
\frac{5^x - 5^{x-1}}{4 \times 5^{x-1}}
\][/tex]
1. Identify the terms in the numerator:
The numerator is [tex]\(5^x - 5^{x-1}\)[/tex].
2. Factor out the common term in the numerator:
Both terms in the numerator share a common factor of [tex]\(5^{x-1}\)[/tex]. So, factor [tex]\(5^{x-1}\)[/tex] out:
[tex]\[
5^x - 5^{x-1} = 5^{x-1} \cdot 5 - 5^{x-1} = 5^{x-1}(5 - 1) = 5^{x-1} \cdot 4
\][/tex]
So the numerator [tex]\(5^x - 5^{x-1}\)[/tex] simplifies to [tex]\(4 \cdot 5^{x-1}\)[/tex].
3. Simplify the fraction:
Substitute the simplified numerator back into the original expression:
[tex]\[
\frac{4 \cdot 5^{x-1}}{4 \times 5^{x-1}}
\][/tex]
4. Cancel the common terms in the numerator and the denominator:
Since the numerator and the denominator are the same, they cancel out:
[tex]\[
\frac{4 \cdot 5^{x-1}}{4 \times 5^{x-1}} = 1
\][/tex]
5. Final simplification:
After canceling the common terms, the simplified expression is:
[tex]\[
1
\][/tex]
So, the simplified form of [tex]\(\frac{5^x - 5^{x-1}}{4 \times 5^{x-1}}\)[/tex] is simply:
[tex]\[
\boxed{1}
\][/tex]
