Question
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Answer
The simplified expression is \(19+26i\).
Solution
Calculate or simplify the expression \( (4+i)(6+5i) \).
Simplify the expression by following steps:
- step0: Solution:
\(\left(4+i\right)\left(6+5i\right)\)
- step1: Apply the distributive property:
\(4\times 6+4\times 5i+i\times 6+i\times 5i\)
- step2: Multiply the numbers:
\(24+4\times 5i+i\times 6+i\times 5i\)
- step3: Multiply the numbers:
\(24+20i+i\times 6+i\times 5i\)
- step4: Multiply the numbers:
\(24+20i+6i+i\times 5i\)
- step5: Multiply the numbers:
\(24+20i+6i-5\)
- step6: Calculate:
\(19+20i+6i\)
- step7: Multiply the numbers:
\(19+26i\)
The simplified expression is \(19+26i\).
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Simplify this solution Beyond the Answer
To simplify the expression \((4+i)(6+5i)\), we can use the distributive property (also known as the FOIL method for binomials). 1. **First Terms**: \(4 \cdot 6 = 24\) 2. **Outer Terms**: \(4 \cdot 5i = 20i\) 3. **Inner Terms**: \(i \cdot 6 = 6i\) 4. **Last Terms**: \(i \cdot 5i = 5i^2\) Since \(i^2 = -1\), we replace \(5i^2\) with \(-5\). Now, we combine all these results: \[ 24 + 20i + 6i - 5 = 24 - 5 + (20i + 6i) \] This simplifies to: \[ 19 + 26i \] Therefore, the simplified expression is: \[ \boxed{19 + 26i} \]
