Question 23

The product (3 + √(- 16))(2 - √(- 25)) is equal to 26 - 7i.

To find the product of the two complex numbers, we first simplify the square roots: √(-16) = 4i and √(-25) = 5i. Substituting these values into the expression yields (3 + 4i)(2 - 5i), which simplifies to 26 - 7i.

A) - 14 + 23i

This option is incorrect as it does not reflect the correct calculations of the product. The real and imaginary parts do not match the outcomes derived from multiplying the complex numbers from the given expression.

B) -14-7i

This choice also does not correspond to the correct result of the multiplication. The real part is incorrect, and while the imaginary part matches the signs, the real component fails to represent the actual outcome.

C) - 7 + 0i

This option is incorrect because the product does not yield a purely real number with no imaginary component. The calculations indicate a significant real part of 26 and an imaginary part of -7i.

D) 5 - i

This choice is incorrect as well. The values derived from the calculations do not support this result, as both the real and imaginary parts differ from what is obtained through the multiplication of the complex numbers.

E) 26-7i

This option is correct. The calculation of the product (3 + 4i)(2 - 5i) results in 26 - 7i after applying the distributive property and combining like terms.

Conclusion

The correct answer is definitively 26 - 7i as it accurately represents the result of the expression when simplified. All other options fail to align with the calculated product, either due to incorrect real or imaginary components or both.