Problem 67 Write each expression in the for... [FREE SOLUTION]

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Chapter 7: Problem 67

Write each expression in the form $a+b i$, where $a$ and $b$ are real numbers. $$2+\sqrt{-4}$$

Short Answer

Expert verified

2 + 2i

Step by step solution

Identify the imaginary component

Recognize that $\sqrt{-4}$ can be rewritten using the imaginary unit $i$, where $i = \sqrt{-1}$.

Simplify the square root

Write $\sqrt{-4}$ as $\sqrt{4} \cdot \sqrt{-1} = 2i$.

Combine real and imaginary parts

Combine the real number $2$ and the imaginary number $2i$ into the form $a + bi$. Thus, $2 + \sqrt{-4} = 2 + 2i$.

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Key Concepts

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

Imaginary Unit

The imaginary unit, denoted as $i$, is a fundamental concept in complex numbers. It is defined as $i = \sqrt{-1}$. This means that $i^2 = -1$.

Imaginary numbers extend the real number system, allowing for the square root of negative numbers, which is not possible with real numbers alone.
For example, the square root of $-4$ can be broken down using the imaginary unit: $\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i$.

Square Roots

When dealing with square roots of negative numbers, imaginary units come into play. In the given problem, we have $ \sqrt{-4} $.

Let's break this down step by step. We know that:

$\sqrt{-4} = \sqrt{4 \cdot -1}$
$\sqrt{4 \cdot -1} = \sqrt{4} \cdot \sqrt{-1}$
$\sqrt{4} = 2$, and since $\sqrt{-1} = i$

Combining these results, we get $\sqrt{-4} = 2i$.

This method allows us to convert square roots of negative numbers into a form involving the imaginary unit $i$, making them easier to work with in expressions.

Combining Real and Imaginary Parts

Combining real and imaginary parts is essential in expressing complex numbers. A complex number is typically written in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit.

In our problem, we have $2 + \sqrt{-4}$. From the previous sections, we know that $\sqrt{-4}$ simplifies to $2i$.

So, combining the real part $2$ and the imaginary part $2i$, we get the complex number $2 + 2i$.

The real part: $2$
The imaginary part: $2i$

This method is widely used in mathematics and engineering to simplify and solve problems involving complex numbers.

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91影视

Write each expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$2+\sqrt{-4}$$

Short Answer

Step by step solution

Identify the imaginary component

Simplify the square root

Combine real and imaginary parts

Key Concepts

Imaginary Unit

Square Roots

Combining Real and Imaginary Parts

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