18. If ab ≠ 0, which of the following is equivalent to (8a * b ^ 7)(- 7a ^ - 5 * b ^ 2) ?
Answer: D
-(56b⁹/a^4)
The expression (8a * b^7)(-7a^-5 * b^2) simplifies to -(56b⁹/a^4), which is the correct equivalent form.
A) b⁹/a^4
This option is incorrect because it lacks the negative sign and the coefficient of 56. The expression derived from multiplying the given terms includes both a negative factor and a product of coefficients that results in 56.
B) -(8b⁹/7a^4)
This option is incorrect as it misrepresents the coefficients of the expression. The correct multiplication yields a coefficient of 56, not 8/7, making this option not equivalent to the original expression.
C) -(56b^14/a^5)
This option is incorrect because it miscalculates the powers of b and a in the final expression. The correct simplification shows that b's power should be 9 and a's power 4, not 14 and 5, respectively.
D) -(56b⁹/a^4)
This option is correct as it accurately reflects the multiplication and simplification of the terms involved. The negative sign, coefficient of 56, and the correct powers of b and a are all properly accounted for.
E) -56a^5b^14
This option is incorrect because it incorrectly places the powers of a and b. The correct expression maintains the powers of b at 9 and a at -4, contrasting with the powers provided in this option.
Conclusion
The correct answer, -(56b⁹/a^4), accurately reflects the product of the two expressions given in the question. All other options fail due to miscalculation of coefficients or incorrect powers of a and b, demonstrating a lack of alignment with the mathematical operations performed. Thus, D is definitively the right choice based on the simplification of the expression.