To find the equivalent expression of \((5^{4} \cdot b^{-10})^{-6}\), we use the rule for exponents that says \((a^m \cdot b^n)^p = a^{m \cdot p} \cdot b^{n \cdot p}\).
Applying this rule to our original expression, we raise each factor inside the parentheses to the power of -6: The factor with base 5: \(5^{4}\) raised to the power of -6 becomes \(5^{4 \cdot -6} = 5^{-24}\). The factor with base b: \(b^{-10}\) raised to the power of -6 becomes \(b^{-10 \cdot -6} = b^{60}\). So now, we have \(5^{-24} \cdot b^{60}\). Given the options available: a. \(5^{4}\cdot b^{60}\) is incorrect because \(5^{4}\) should have been raised to the power of -6. b. \(5^{24}\cdot b^{60}\) is incorrect because the exponent of 5 should be -24, not 24.
c. \(\dfrac{b^{60}}{5^{24}}\) is incorrect because the exponent of 5 in the denominator should be -24, which would put it in the numerator when simplified. d. None of the above Therefore, the correct answer is d. None of the above. The true equivalent expression is not listed in the options. It is \(5^{-24} \cdot b^{60}\).