To find an expression equivalent to \([(3+2i) \cdot (4i)] \cdot (2+5i)\), we can analyze the given options:
- Option 1: \([(12i + 8) \cdot (2 + 5i)]\)
- Option 2: \((3 + 2i) \cdot [4i + 2 + 5i]\)
- Option 3: \((3 + 2i) \cdot [(4i) \cdot (2 + 5i)]\)
- Option 4: \((3 + 2i) + [(4i) + (2 + 5i)]\)
Let's break down the original expression step by step:
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Calculate \((3 + 2i) \cdot (4i)\):
- \[ (3 + 2i)(4i) = 3 \cdot 4i + 2i \cdot 4i = 12i + 8i^2 \]
- Since \(i^2 = -1\), it becomes:
- \[ 12i - 8 = -8 + 12i \]
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Now, take this result and multiply it by \((2 + 5i)\):
- \[ (-8 + 12i)(2 + 5i) \]
Next, let's check the options:
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Option 1 involves multiplying \((12i + 8)\) by \((2 + 5i)\). This is not correct because the multiplication should involve \((-8 + 12i)\), not \((12i + 8)\).
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Option 2 is \((3 + 2i) \cdot (4i + 2 + 5i)\), which rearranges terms from the first multiplication, but it does not follow the correct order of operations.
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Option 3 is \((3 + 2i) \cdot [(4i) \cdot (2 + 5i)]\). This keeps the correct initial multiplication—the nested multiplication of \((4i)\) and \((2 + 5i)\) would yield a valid expression, and applying it gives the equivalent result.
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Option 4 involves addition, which does not match the original expression structure.
From the analysis, Option 3, \((3 + 2i) \cdot [(4i) \cdot (2 + 5i)]\), is the correct one as it maintains the required producing equivalent results through the correct order of operations.
Therefore, the correct choice is:
\((3 + 2i) \cdot [(4i) \cdot (2 + 5i)]\) (Option 3).
1 answer