To solve the given expression [tex]\(\frac{6^{-10}}{6^4 \cdot 6^0}\)[/tex] by applying the laws of exponents, we can follow these steps:
1. Understand the Expression: The expression given is [tex]\(\frac{6^{-10}}{6^4 \cdot 6^0}\)[/tex].
2. Simplify the Denominator: According to the exponent law [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex], we can add the exponents in the denominator:
[tex]\[
6^4 \cdot 6^0 = 6^{4+0} = 6^4
\][/tex]
So, the expression becomes:
[tex]\[
\frac{6^{-10}}{6^4}
\][/tex]
3. Apply the Quotient of Powers Rule: According to the exponent law [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex], we subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[
\frac{6^{-10}}{6^4} = 6^{-10-4} = 6^{-14}
\][/tex]
4. Rewrite Negative Exponents: According to the exponent law [tex]\(a^{-n} = \frac{1}{a^n}\)[/tex], a negative exponent can be rewritten as the reciprocal of the base with the positive exponent:
[tex]\[
6^{-14} = \frac{1}{6^{14}}
\][/tex]
Therefore, the expression equivalent to [tex]\(\frac{6^{-10}}{6^4 \cdot 6^0}\)[/tex] is [tex]\(\frac{1}{6^{14}}\)[/tex].
Among the given choices, the correct answer is:
[tex]\[
\boxed{\frac{1}{6^{14}}}
\][/tex]
