Sure, let's simplify the given expressions step by step:
### Simplification of [tex]\(\frac{a^5}{a^{11}}\)[/tex]:
1. Identify the exponent rule for division: When you divide like bases, you subtract the exponents. The rule is [tex]\( a^m / a^n = a^{m-n} \)[/tex].
2. Apply the rule: Here, [tex]\( m = 5 \)[/tex] and [tex]\( n = 11 \)[/tex].
[tex]\[
\frac{a^5}{a^{11}} = a^{5-11}
\][/tex]
3. Calculate the exponent:
[tex]\[
5 - 11 = -6
\][/tex]
4. Write the result:
[tex]\[
a^{-6}
\][/tex]
So, [tex]\(\frac{a^5}{a^{11}} = a^{-6}\)[/tex].
### Simplification of [tex]\(\frac{5^{11}}{5^{12}}\)[/tex]:
1. Identify the exponent rule for division: When you divide like bases, you subtract the exponents. The rule is [tex]\( 5^m / 5^n = 5^{m-n} \)[/tex].
2. Apply the rule: Here, [tex]\( m = 11 \)[/tex] and [tex]\( n = 12 \)[/tex].
[tex]\[
\frac{5^{11}}{5^{12}} = 5^{11-12}
\][/tex]
3. Calculate the exponent:
[tex]\[
11 - 12 = -1
\][/tex]
4. Write the result:
[tex]\[
5^{-1}
\][/tex]
So, [tex]\(\frac{5^{11}}{5^{12}} = 5^{-1}\)[/tex].
### Summary of the Solutions:
[tex]\[
\frac{a^5}{a^{11}} = a^{-6}
\][/tex]
[tex]\[
\frac{5^{11}}{5^{12}} = 5^{-1}
\][/tex]
