To solve the expression [tex]\(5^5 \div 5^{-6}\)[/tex], we need to use the properties of exponents. Specifically, the rule that states:
[tex]\[
\frac{a^m}{a^n} = a^{m-n}
\][/tex]
This rule tells us that when we divide powers with the same base, we subtract the exponent of the denominator from the exponent of the numerator.
Given the expression [tex]\(5^5 \div 5^{-6}\)[/tex]:
1. Identify the base (which is 5).
2. Identify the exponents (5 for the numerator and -6 for the denominator).
3. Apply the exponent rule [tex]\(a^m / a^n = a^{m-n}\)[/tex].
So, we need to subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[
5^5 \div 5^{-6} = 5^{5 - (-6)}
\][/tex]
Subtracting a negative exponent is the same as adding its positive counterpart:
[tex]\[
5^{5 - (-6)} = 5^{5 + 6}
\][/tex]
Now, add the exponents:
[tex]\[
5^{5 + 6} = 5^{11}
\][/tex]
Therefore, the value of [tex]\(5^5 \div 5^{-6}\)[/tex] is [tex]\(5^{11}\)[/tex].
Hence, the answer is:
c. [tex]\(5^{11}\)[/tex]
