Ch. 1 - Equations and Inequalities

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 1 - Equations and Inequalities

Problem 68a

Chapter 2, Problem 68a

Find each product. Write answers in standard form. i(2+7i)(2-7i)

Verified step by step guidance

First, recognize that the expression involves the product of three factors: $i$, $(2 + 7i)$, and $(2 - 7i)$.

Next, multiply the two complex conjugates $(2 + 7i)$ and $(2 - 7i)$ using the difference of squares formula: $(a + bi)(a - bi) = a^2 + b^2$. So, calculate $2^2 + 7^2$.

Simplify the result from the previous step to get a real number.

Now multiply this real number by $i$ to get the final product.

Express the final answer in standard form $a + bi$, where $a$ and $b$ are real numbers.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Complex Numbers and Imaginary Unit

Complex numbers have 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 manipulate i is essential for working with expressions involving complex numbers.

Multiplication of Complex Numbers

Multiplying complex numbers involves using the distributive property (FOIL method) and combining like terms, remembering that i² = -1. This process allows simplification of products into standard form a + bi.

Difference of Squares Formula

The expression (a + bi)(a - bi) uses the difference of squares formula: (x + y)(x - y) = x² - y². Applying this to complex conjugates simplifies the product to a real number, which is useful in simplifying complex expressions.

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