Question
Multiply. \( (-4+9 i)(8-5 i) \) \( (-4+9 i)(8-5 i)=\square \) (Simplify your answer. Type your answer in the form a + bi.)
Ask by Schneider Ortega. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( 13 + 92i \)
Solution
Calculate or simplify the expression \( (-4+9i)(8-5i) \).
Simplify the expression by following steps:
- step0: Solution:
\(\left(-4+9i\right)\left(8-5i\right)\)
- step1: Apply the distributive property:
\(-4\times 8-4\left(-5i\right)+9i\times 8+9i\left(-5i\right)\)
- step2: Multiply the numbers:
\(-32-4\left(-5i\right)+9i\times 8+9i\left(-5i\right)\)
- step3: Multiply the numbers:
\(-32+20i+9i\times 8+9i\left(-5i\right)\)
- step4: Multiply the numbers:
\(-32+20i+72i+9i\left(-5i\right)\)
- step5: Multiply the numbers:
\(-32+20i+72i+45\)
- step6: Calculate:
\(13+20i+72i\)
- step7: Multiply the numbers:
\(13+92i\)
The product of \( (-4+9i)(8-5i) \) is \( 13+92i \).
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Simplify this solution Bonus Knowledge
To multiply the complex numbers \((-4 + 9i)\) and \((8 - 5i)\), we use the distributive property (also known as the FOIL method for binomials): \[ (-4)(8) + (-4)(-5i) + (9i)(8) + (9i)(-5i) \] Calculating each term: 1. \((-4)(8) = -32\) 2. \((-4)(-5i) = 20i\) 3. \((9i)(8) = 72i\) 4. \((9i)(-5i) = -45i^2\) (Since \(i^2 = -1\), this becomes \(+45\)) Now, combine all these results: \[ -32 + 20i + 72i + 45 \] Combining like terms: 1. Real parts: \(-32 + 45 = 13\) 2. Imaginary parts: \(20i + 72i = 92i\) Thus, the final result is: \[ (-4 + 9i)(8 - 5i) = 13 + 92i \] So, the answer is \( \boxed{13 + 92i} \).
