Problem 24 Express in terms of $i$ $$ ... [FREE SOLUTION]

Chapter 10: Problem 24

Express in terms of $i$ $$ \sqrt{-4}+\sqrt{-12} $$

Short Answer

Expert verified

鈭�-4 + 鈭�-12 = 2i + 2i鈭�3

Step by step solution

Understand the Problem

Realize that you need to simplify the expression involving square roots of negative numbers. This can be done by expressing the square roots in terms of the imaginary unit, denoted by $i$.

Express \sqrt{-4}\ in Terms of $i$

Rewrite \sqrt{-4}\ as \sqrt{4} \times \sqrt{-1}\. Then, using the fact that \[ \sqrt{-1} = i \], we can simplify it to: \[ \sqrt{-4} = 2i \]

Express \sqrt{-12}\ in Terms of $i$

Rewrite \sqrt{-12}\ as \sqrt{12} \times \sqrt{-1}\. Knowing that \[ \sqrt{12} = 2 \sqrt{3} \], we have: \[ \sqrt{-12} = 2 \sqrt{3} \times i = 2i \sqrt{3} \]

Add the Simplified Expressions

Combine the simplified expressions from Steps 2 and 3 as follows: \[ 2i + 2i \sqrt{3} \]

Simplify the Final Expression

The final simplified expression is: \[ \sqrt{-4} + \sqrt{-12} = 2i + 2i \sqrt{3} \]

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Key Concepts

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

square roots of negative numbers

When dealing with the square roots of negative numbers, we encounter an interesting mathematical concept. Usually, taking the square root of a positive number gives us a real number. But if the number inside the square root is negative, things change. For example, we might need to find $\root{-4}$. Since there is no real number that, when squared, gives a negative result, we turn to imaginary numbers. This is where the imaginary unit $i$ comes into play.

imaginary unit (i)

The imaginary unit, denoted as $i$, is defined by the equation $i = \root{-1}$. This definition allows us to express the square roots of negative numbers in a simplified form. For instance:

$\root{-4} = \root{4} \times \root{-1}$
As $i = \root{-1}$, this can be rewritten as $2i$

We use this property of $i$ to handle square roots of other negative numbers too. Let's take \root{-12} as an example.

simplifying radical expressions

Simplifying expressions involving radicals, especially ones with negative numbers, requires using the imaginary unit carefully. To simplify $ \root{-12}$, we can first break it down:

$ \root{12} \times \root{-1} $
Recognizing $ \root{12} = 2 \root{3}$, we can write: $ \root{-12} = 2 \root{3} \times i = 2i \root{3} $

Finally, we can add our simplified expressions from the problem: $ \root{-4} + \root{-12} = 2i + 2i \root{3} $. Following these steps helps to bring clarity and understanding to working with imaginary numbers and radical expressions.

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91影视

Express in terms of \(i\) $$ \sqrt{-4}+\sqrt{-12} $$

Short Answer

Step by step solution

Understand the Problem

Express \sqrt{-4}\ in Terms of \(i\)

Express \sqrt{-12}\ in Terms of \(i\)

Add the Simplified Expressions

Simplify the Final Expression

Key Concepts

square roots of negative numbers

imaginary unit (i)

simplifying radical expressions

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