Textbook Question
Find each product. Write answers in standard form. i(3-4i)(3+4i)
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1. Equations & Inequalities
Complex Numbers
1:53 minutesProblem 81a
Textbook Question
Find each quotient. Write answers in standard form. 8 / -i
Verified step by step guidance1
Recall that dividing by a complex number can be simplified by multiplying the numerator and denominator by the complex conjugate of the denominator. Here, the denominator is \(-i\), and its conjugate is \(i\).
Multiply both the numerator and denominator by \(i\) to eliminate the imaginary unit from the denominator: \(\frac{8}{-i} \times \frac{i}{i} = \frac{8i}{-i \cdot i}\).
Simplify the denominator using the fact that \(i^2 = -1\): \(-i \cdot i = -i^2 = -(-1) = 1\).
Now the expression becomes \(\frac{8i}{1}\), which simplifies to \$8i$.
Write the answer in standard form \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(a = 0\) and \(b = 8\), so the answer is \$0 + 8i$.
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Complex Numbers and the Imaginary Unit
Complex numbers consist of a real part and an imaginary part, expressed as a + bi, where i is the imaginary unit with the property i² = -1. Understanding how to work with i is essential for simplifying expressions involving complex numbers.
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Introduction to Complex Numbers
Division of Complex Numbers
Dividing complex numbers often involves multiplying the numerator and denominator by the conjugate of the denominator to eliminate imaginary terms from the denominator. This process simplifies the expression into standard form a + bi.
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Standard Form of Complex Numbers
The standard form of a complex number is a + bi, where a and b are real numbers. Writing answers in this form means separating the real and imaginary parts clearly, which is the goal when simplifying quotients involving complex numbers.
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