((5^3 * 2^4)^2)(5^(-2) * 2^5)

Rationale

Extracting the expression results in 5^4 * 2^13.

When simplifying the given expression ((5^3 * 2^4)^2)(5^(-2) * 2^5), we first apply the power of a product rule and then combine like bases to arrive at the final simplified form of 5^4 * 2^13.

A (5^3 * 2^11) YOUR ANSWER

This choice results from an incorrect simplification of the expression. The calculation mistakenly does not account for the squaring of both bases in the first term or the full combination of exponents from both parts of the expression.

B (5^(-12) * 2^40) YOUR ANSWER

This option represents an erroneous outcome due to miscalculating the exponents. The negative exponent for the base of 5 suggests a misunderstanding of the multiplication and simplification process, as the correct operations yield positive exponents.

C (5^4 * 2^13) CORRECT

This is the correct choice, as it properly reflects the result of applying the rules of exponents. Simplifying the expression yields 5^(3*2 - 2) = 5^4 and 2^(4*2 + 5) = 2^13, confirming the accuracy of the calculation.

D ((-5)^8 * 2^13) YOUR ANSWER

This choice incorrectly incorporates a negative base in the power of 5. The original expression does not involve a negative component, so the transition to (-5)^8 is not justified and does not align with the operations performed.

Conclusion

Upon simplification of the expression ((5^3 * 2^4)^2)(5^(-2) * 2^5), we find that the correct outcome is 5^4 * 2^13. Understanding the laws of exponents is crucial in obtaining the correct answer, as demonstrated by the incorrect answers that arise from misapplication of these rules.

((5^3 * 2^4)^2)(5^(-2) * 2^5)

Extracting the expression results in 5^4 * 2^13.

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