To simplify the expression \( 3i[(2i) + (-3i - 5)] \) using the Distributive Property, let's follow these steps:
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Simplify the expression inside the brackets: \[ (2i) + (-3i - 5) = 2i - 3i - 5 = -i - 5 \]
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Now substitute back into the original expression: \[ 3i[-i - 5] \]
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Use the Distributive Property to multiply \( 3i \) with each term in the parentheses: \[ = 3i \cdot (-i) + 3i \cdot (-5) \]
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Calculate each term:
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For \( 3i \cdot (-i) \): \[ 3i \cdot (-i) = -3i^2 \] Since \( i^2 = -1 \), this becomes: \[ -3(-1) = 3 \]
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For \( 3i \cdot (-5) \): \[ 3i \cdot (-5) = -15i \]
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Combine the results: \[ 3 - 15i \]
Therefore, the simplified form of \( 3i[(2i) + (-3i - 5)] \) is:
\[ \boxed{3 - 15i} \]
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