Simplify (3 − 4i)(3 + 4i).

Rationale

Extracting the product of (3 − 4i)(3 + 4i) results in 25.

The expression represents the product of two conjugate complex numbers, which simplifies to a real number through the formula ( (a - bi)(a + bi) = a^2 + b^2 ). Here, ( a = 3 ) and ( b = 4 ), leading to a final result of 25.

A (−25) YOUR ANSWER

This option suggests a negative outcome, which is incorrect. The product of the two conjugate complex numbers is positive because the squares of both real and imaginary parts (3 and 4) yield positive results. Thus, the calculation will not result in a negative number.

B (−7) YOUR ANSWER

This choice implies a negative real number as the result, which does not align with the calculation of the product. The operation involves squaring the components, which leads to a positive sum, ruling out any possibility of a negative outcome.

C (0) YOUR ANSWER

Choosing 0 suggests that the product of these two complex numbers cancels out, which is not the case. Conjugate pairs multiply to produce a positive real number, specifically the sum of the squares of their respective components, confirming the result cannot be zero.

D (25) CORRECT

The product simplifies accurately to 25, as calculated by ( 3^2 + 4^2 = 9 + 16 = 25 ). This is the correct answer, demonstrating that the operation of multiplying the conjugate pairs yields a positive real number.

E (7) YOUR ANSWER

This option proposes a small positive number, which underestimates the results of the calculations. The sum of the squares of 3 and 4 yields a much larger result, confirming that 7 is not a valid simplification of the expression.

Conclusion

The simplification of the expression (3 − 4i)(3 + 4i) correctly yields 25, as derived from the formula for multiplying complex conjugates. Each incorrect choice misrepresents the outcome, either suggesting negative results or underestimating the positive value obtained from the operation. Understanding the nature of complex conjugates is crucial, as it illuminates the relationship between real and imaginary components in multiplication.

Simplify (3 − 4i)(3 + 4i).

Extracting the product of (3 − 4i)(3 + 4i) results in 25.

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