Problem 9 Express in terms of $i$ $$ ... [FREE SOLUTION]

Chapter 10: Problem 9

Express in terms of $i$ $$ \sqrt{-100} $$

Short Answer

Expert verified

$10i$

Step by step solution

Understand Complex Numbers

When dealing with square roots of negative numbers, convert them into terms of the imaginary unit, denoted as $i$. Remember, $i$ is defined as $i = \sqrt{-1}$.

Factor the Negative Sign

Rewrite the original expression $\sqrt{-100}$ by factoring the negative sign. This converts the problem into $\sqrt{-1 \times 100}$.

Apply the Square Root Property

Use the property of square roots to separate the terms: $\sqrt{-1 \times 100} = \sqrt{-1} \times \sqrt{100}$.

Simplify the Individual Square Roots

Simplify $\sqrt{-1}$ as $i$ (since $i = \sqrt{-1}$) and $\sqrt{100}$ as 10. This leads to $i \times 10$.

Combine the Results

Combine the simplified terms to get the final result: $10i$.

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Key Concepts

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

imaginary unit

When you encounter the square root of a negative number, it often looks confusing. This is where the imaginary unit comes into play. The imaginary unit is denoted as $i$ and is defined by just one important property:
\[i = \sqrt{-1}\].
This simple definition transforms the way we handle negative square roots. If you need to find the square root of any negative number, you will use this property.

square root property

The square root property is a guiding tool for dealing with complex numbers. It states that the square root of a product is equal to the product of the square roots. This can be written as:
\[ \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \].
For example, to solve $ \sqrt{-100} $, we rewrite it by separating the negative sign and applying the square root property:
\sqrt{-100} = \sqrt{-1 \cdot 100} = \sqrt{-1} \cdot \sqrt{100} = i \cdot 10 = 10i.
This property helps in simplifying expressions and solving equations.

simplifying radical expressions

Simplifying radical expressions is the step-by-step process that resolves complex numbers into simpler forms. Here is how you would simplify $ \sqrt{-100} $:

Identify the negative inside the square root.
Rewrite it in terms of $i$ using the imaginary unit ($i = \sqrt{-1}$).
Separate using the square root property ( $ \sqrt{-100} = \sqrt{-1 \cdot 100} $ ).
Simplify each part ( $ \sqrt{-1} = i $ and $ \sqrt{100} = 10 $ ).
Combine for the final result: $ 10i $.

This process ensures that all negative square roots are handled properly and your final result is clear and accurate.

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91影视

Express in terms of \(i\) $$ \sqrt{-100} $$

Short Answer

Step by step solution

Understand Complex Numbers

Factor the Negative Sign

Apply the Square Root Property

Simplify the Individual Square Roots

Combine the Results

Key Concepts

imaginary unit

square root property

simplifying radical expressions

One App. One Place for Learning.

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