Problem 45 Find the product of the given co... [FREE SOLUTION]

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Chapter 7: Problem 45

Find the product of the given complex number and its conjugate. $$3+5 i$$

Short Answer

Expert verified

The product is 34.

Step by step solution

Identify the Complex Number and Its Conjugate

The given complex number is $3+5i$. The complex conjugate of $3+5i$ is $3-5i$.

Multiply the Complex Number and Its Conjugate

To find the product, multiply $3+5i$ by $3-5i$: \[(3+5i)(3-5i)\]

Use the Difference of Squares Formula

Use the formula $(a+b)(a-b) = a^2 - b^2$, where $a = 3$ and $b = 5i$: \[(3+5i)(3-5i) = 3^2 - (5i)^2\]

Simplify the Expression

Calculate the squares: \[3^2 - (5i)^2 = 9 - 25i^2\]Since $i^2 = -1$, substitute $-1$ for $i^2$: \[9 - 25(-1) = 9 + 25\]Therefore: \[9 + 25 = 34\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

complex conjugate

A complex conjugate is a fundamental concept in complex numbers. It helps in simplifying products and quotients of complex numbers. Given a complex number of the form 3 + 5i (where i is the imaginary unit), its conjugate is the same number with the sign of the imaginary part changed. So, the conjugate of 3 + 5i is 3 - 5i.

The utility of the complex conjugate comes into play when you are multiplying complex numbers. For instance, multiplying a complex number by its conjugate simplifies calculations because we use the difference of squares formula.

difference of squares

The difference of squares formula is a key algebraic tool: (a + b)(a - b) = a^2 - b^2. This formula is useful when dealing with complex numbers. For example, when multiplying a complex number 3 + 5i by its conjugate 3 - 5i, you can apply the difference of squares formula.

Here, a = 3 and b = 5i, so:
(3 + 5i)(3 - 5i) = 3^2 - (5i)^2.
Calculating the squares gives us 3^2 = 9 and (5i)^2 = 25i^2. Since i^2 = -1, (5i)^2 becomes 25(-1) or -25. Thus, 9 - (-25) turns into 9 + 25 = 34.

imaginary unit

The imaginary unit, denoted as i, is the square root of -1. It's a fundamental aspect of complex numbers. The property that i^2 = -1 is crucial in simplifying expressions involving complex numbers.

When dealing with complex conjugates, the imaginary unit helps simplify calculations. For instance, in the expression (3 + 5i)(3 - 5i), we encounter the term (5i)^2. Knowing that i^2 = -1, we can simplify (5i)^2 to 25(-1) or -25. This simplification highlights the role of the imaginary unit in handling complex numbers efficiently.

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91影视

Find the product of the given complex number and its conjugate. $$3+5 i$$

Short Answer

Step by step solution

Identify the Complex Number and Its Conjugate

Multiply the Complex Number and Its Conjugate

Use the Difference of Squares Formula

Simplify the Expression

Key Concepts

complex conjugate

difference of squares

imaginary unit

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