Problem 23 Write in terms of $i$ and then... [FREE SOLUTION]

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Trigonometry

Charles P. McKeague, Mark D. Turner

$Math Studyset 91影视 Explanations$ Math

8 Edition

Chapter 8: Problem 23

Write in terms of $i$ and then simplify. $$\sqrt{-4} \cdot \sqrt{-9}$$

Short Answer

Expert verified

The expression simplifies to -6.

Step by step solution

Express the Square Roots in Terms of i

Recall that the square root of a negative number can be expressed in terms of the imaginary unit, $i$, where $i = \sqrt{-1}$. Therefore, $\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i$ and $\sqrt{-9} = \sqrt{9} \cdot \sqrt{-1} = 3i$.

Multiply the Expressions

Now that both square roots are expressed in terms of $i$, we can multiply them together: $ (2i) \cdot (3i) = 6i^2 $.

Simplify Using the Property of i^2

Recall that $i^2 = -1$. We can simplify the expression $6i^2$ by substitute $i^2$ with $-1$: $6 \cdot (-1) = -6$.

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Key Concepts

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

Imaginary Unit

When dealing with complex numbers, the imaginary unit plays a critical role. The symbol $i$ is used to denote the imaginary unit, and it is defined as $i = \sqrt{-1}$. Why do we need the imaginary unit? Because in the real number system, we cannot take the square root of a negative number. However, by introducing $i$, we can extend our number system to include these numbers, which we call complex numbers. Complex numbers have the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit.

**$i$** allows us to handle square roots of negative numbers, turning these challenging numbers into a more manageable form.
The property $i^2 = -1$ is fundamental in simplifying expressions involving $i$.

Square Roots of Negative Numbers

Square roots of negative numbers can often appear complex and confusing. However, they become simpler when using the imaginary unit $i$. For instance, if you need to find $\sqrt{-4}$, you can rewrite it using the properties of square roots combined with $i$:

Express $\sqrt{-4}$ as $\sqrt{4} \times \sqrt{-1}$, which becomes $2i$.
Similarly, $\sqrt{-9}$ can be rewritten as $3i$, using $\sqrt{9} \times \sqrt{-1}$.

By expressing square roots of negative numbers in this way, calculations become feasible and align with the rules of arithmetic.
This approach aids in performing multiplications, divisions, and other operations with complex numbers without violating any mathematical rules.

Simplification of Expressions

Simplifying complex number expressions, especially those involving $i$, can be straightforward if you follow a few basic steps. Let's take a closer look at the process used in the original problem $\sqrt{-4} \cdot \sqrt{-9}$:

First, express each square root in terms of $i$: $\sqrt{-4} = 2i$ and $\sqrt{-9} = 3i$.
Next, multiply the resulting expressions: $(2i) \cdot (3i) = 6i^2$.
Finally, use the property of $i$ where $i^2 = -1$ to simplify further: $6i^2 = 6 \times -1 = -6$.

By following these steps, you transform complex expressions into simpler, manageable forms. Practicing this process with different examples helps solidify the understanding needed to solve problems involving complex numbers with confidence.

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91影视

Write in terms of \(i\) and then simplify. $$\sqrt{-4} \cdot \sqrt{-9}$$

Short Answer

Step by step solution

Express the Square Roots in Terms of i

Multiply the Expressions

Simplify Using the Property of i^2

Key Concepts

Imaginary Unit

Square Roots of Negative Numbers

Simplification of Expressions

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