Express the complex number expression in simplest form where...

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--- ## Question 1 **Question content**: Express the complex number expression \( 3yi^3 + 7yi^4 \) in simplest form, where \( i \) is the imaginary unit (\( i = \sqrt{-1} \)). **Discipline**: Mathematics **Grade**: High School (typically taught in Algebra II or Precalculus, around 10th-12th grade) ### Correct answer \( 7y - 3yi \) (or equivalently \( y(7 - 3i) \)) ### Detailed problem-solving steps 1. **Recall the powers of the imaginary unit \( i \)**: By definition, \( i = \sqrt{-1} \), so: - \( i^2 = (\sqrt{-1})^2 = -1 \) - \( i^3 = i^2 \cdot i = -1 \cdot i = -i \) - \( i^4 = (i^2)^2 = (-1)^2 = 1 \) 2. **Substitute \( i^3 \) and \( i^4 \) into the expression**: - For the first term \( 3yi^3 \): Substitute \( i^3 = -i \): \( 3yi^3 = 3y(-i) = -3yi \) - For the second term \( 7yi^4 \): Substitute \( i^4 = 1 \): \( 7yi^4 = 7y(1) = 7y \) 3. **Combine the simplified terms**: Add the two simplified terms: \( 3yi^3 + 7yi^4 = -3yi + 7y \) Rearrange into standard complex number form (\( \text{real part} + \text{imaginary part} \)): \( 7y - 3yi \) (or factor \( y \): \( y(7 - 3i) \)) ### Knowledge points involved 1. **Imaginary Unit (\( i \))** - Definition: \( i = \sqrt{-1} \), introduced to extend the real number system to complex numbers (where solutions to equations like \( x^2 = -1 \) exist). - Basic power: \( i^2 = -1 \) (a fundamental property used to simplify higher powers of \( i \)). 2. **Powers of \( i \) (Cyclic Pattern)** - Higher powers of \( i \) follow a cyclic pattern with period 4: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), \( i^4 = 1 \), \( i^5 = i \), \( i^6 = -1 \), etc. - This cyclicity comes from repeatedly multiplying by \( i \) (or using \( i^2 = -1 \) to simplify). 3. **Simplifying Complex Number Expressions** - Complex numbers have the form \( a + bi \) (real part \( a \), imaginary part \( b \)). - To simplify, substitute powers of \( i \) with their equivalent forms (using \( i^2 = -1 \), \( i^3 = -i \), \( i^4 = 1 \), etc.), then combine like terms (real and imaginary parts). ---

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