Problem 81 Simplify. Write each result in a... [FREE SOLUTION]

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Beginning Algebra: A Guided Approach

Rosemary Karr, Marilyn Massey, R. David Gustafson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 9: Problem 81

Simplify. Write each result in a + bi form. $$(2+\sqrt{-3})(3-\sqrt{-2})$$

Short Answer

Expert verified

The simplified answer is $6+\sqrt{6}$ + (3$\sqrt{3}$-2$\sqrt{2}$)i.

Step by step solution

Identify the Complex Numbers

The two complex numbers are (2+$i\sqrt{3}$) and (3-$i\sqrt{2}$). The $i\sqrt{-3}$ and $i\sqrt{-2}$ are imaginary numbers since sqrt(-3) and sqrt(-2) is not defined in the real number system but is in the complex number system. Here, the 'i' represents $\sqrt{-1}$.

Multiply the Complex Numbers

To multiply the numbers, apply the distribution property. Multiply each term of the first complex number by each term of the second complex number. This yields 3*2 + 3*$i\sqrt{3}$ - 2*$i\sqrt{2}$ - $i\sqrt{3}$*$i\sqrt{2}$. This simplifies to 6+3$i\sqrt{3}$-2$i\sqrt{2}$-$i^{2}\sqrt{6}$.

Simplify the Result

Here, replace $i^{2}$ with -1, since $i^{2}=-1$. So, the equation becomes 6+3$i\sqrt{3}$-2$i\sqrt{2}$+$\sqrt{6}$, which simplifies to $6+\sqrt{6}$+$3i\sqrt{3}-2i\sqrt{2}$. Now group the real and imaginary parts separately, the final solution is $6+\sqrt{6}$ + (3$\sqrt{3}$-2$\sqrt{2}$)i. Thus, the result is in 'a + bi' form.

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Key Concepts

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

Imaginary Numbers

Imaginary numbers are a significant component of complex numbers and are essential for performing calculations involving square roots of negative numbers. They are defined using the imaginary unit 'i', where $ i^2 = -1 $. This means that $ i $ itself is equal to $ \sqrt{-1} $. Imaginary numbers allow us to work with the square roots of negative numbers in a similar way that positive square roots work in the real number system. For instance:

$ \sqrt{-3} $ can be expressed as $ i\sqrt{3} $
$ \sqrt{-2} $ can be expressed as $ i\sqrt{2} $

In our original exercise, we deal with these imaginary parts to form complex numbers, written as $ a + bi $ where $ a $ is the real part and $ bi $ is the imaginary part.

Multiplying Complex Numbers

Multiplying complex numbers involves a process similar to multiplying binomials. You apply the distributive property, often referred to as FOIL (First, Outside, Inside, Last) in this context for the multiplication of two binomials. When multiplying, you multiply each term in the first complex number by each term in the second complex number. For example, when multiplying $(2+i\sqrt{3})(3-i\sqrt{2})$, you would perform the following calculations:

First: $2 \times 3 = 6$
Outside: $2 \times (-i\sqrt{2}) = -2i\sqrt{2}$
Inside: $i\sqrt{3} \times 3 = 3i\sqrt{3}$
Last: $i\sqrt{3} \times (-i\sqrt{2}) = -i^2\sqrt{6}$

Using these steps, you combine results, keeping in mind that $i^2 = -1$, which results in simplifying terms further. The final results need to be combined into their real and imaginary components.

Simplifying Expressions

Simplifying an expression involving complex numbers is the final step in bringing it to the standard form $ a + bi $. After multiplying complex numbers, you often end up with terms that need simplification. This includes the use of the identity $ i^2 = -1 $ to convert complex conjugate terms into real numbers.
In the solution provided, the term $-i^2\sqrt{6}$ converts to $\sqrt{6}$ due to $-i^2$ equating to $1$. Then, carefully combine the real parts and the imaginary parts separately:

Real Parts: $6 + \sqrt{6}$
Imaginary Parts: $3\sqrt{3}i - 2\sqrt{2}i$

By grouping these results together, the expression is simplified into a neat $ a + bi $ form, making it more straightforward for interpretation and further mathematical operations.

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91影视

Simplify. Write each result in a + bi form. $$(2+\sqrt{-3})(3-\sqrt{-2})$$

Short Answer

Step by step solution

Identify the Complex Numbers

Multiply the Complex Numbers

Simplify the Result

Key Concepts

Imaginary Numbers

Multiplying Complex Numbers

Simplifying Expressions

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