Problem 15 Write each expression in terms o... [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 15

Write each expression in terms of $i$. $$\sqrt{-16}$$

Short Answer

Expert verified

The expression $\sqrt{-16}$ can be written as $4i$.

Step by step solution

Understand Complex Numbers

A complex number is a number in the form of $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit. The imaginary unit $i$ has the property that $i^2 = -1$. This exercise requires us to express the given number involving a square root of a negative number using $i$.

Apply the Definition of Imaginary Unit

We start by recognizing that $\sqrt{-1} = i$. The given problem is to find $\sqrt{-16}$, which can be expressed as the product $\sqrt{-1} \times \sqrt{16}$.

Simplify the Expression

We know that $\sqrt{16} = 4$. Therefore, $\sqrt{-16} = \sqrt{-1} \times \sqrt{16} = i \times 4$.

Multiply to Write in Standard Form

Multiply the known values: $4 imes i = 4i$. Therefore, $\sqrt{-16}$ in the form of $a + bi$ becomes $0 + 4i$, which simplifies to $4i$.

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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, represented as $i$, is a fundamental concept in complex numbers. Although the term 'imaginary' might sound like these numbers don't exist, they play a crucial role in mathematics.

The imaginary unit is defined by its unique property: $i^2 = -1$.
This definition is manipulated to help solve equations where square roots of negative numbers appear.

Face a situation in mathematics where you need a solution involving the square root of a negative number? That's when $i$ comes in handy! By defining $i$, mathematicians expanded the number system to include solutions that were not previously possible. It's like opening a new door in a house built on the solid foundation of real numbers, leading to a richer understanding of algebra and beyond.

Square Roots of Negative Numbers

Square roots, in general, are numbers which produce a specified product when multiplied by themselves. However, when it comes to negative numbers, things become a bit more challenging.

Historically, square roots of negative numbers were considered impossible since no real number squared gives a negative result.
With the introduction of the imaginary unit, we can now compute and understand the square roots of negative numbers.

For instance, think about $\sqrt{-16}$. You can break this down using the property of $i$:

First, recognize that $-16$ can be split into $-1\times 16$.
The square root of $-16$ then becomes $\sqrt{-1}\times \sqrt{16}$.
Thanks to $i$, you know $\sqrt{-1} = i$, thus transforming $\sqrt{-16}$ into $4i$, since $\sqrt{16} = 4$.

Such breakdowns help us work with negative roots effortlessly, bringing them into the realm of complex numbers.

Expressions Involving i

Once familiar with $i$, you can construct expressions involving this unit to deal with complex problems. These expressions use $i$ to express numbers that include imaginary parts.

Typically, complex numbers are written in the form $a + bi$, where $a$ is the real part, and $bi$ is the imaginary part.
In our example, $\sqrt{-16}$, the result simplifies to $4i$. Here, $a = 0$ and $b = 4$.

Operations with $i$ follow specific rules, all grounded in the property $i^2 = -1$. Multiplying, adding, and even exponentiating these numbers requires special attention.For any given expression with $i$:

Combine like terms, just like you would with algebraic expressions.
Be mindful of multiplying $i$ because $i \times i = i^2 = -1$.
Simplify to achieve the typical form of a complex number.

Exploring expressions with $i$ showcases the elegance of complex numbers and enriches your analytical skills in handling mathematical problems.

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91影视

Write each expression in terms of \(i\). $$\sqrt{-16}$$

Short Answer

Step by step solution

Understand Complex Numbers

Apply the Definition of Imaginary Unit

Simplify the Expression

Multiply to Write in Standard Form

Key Concepts

Imaginary Unit

Square Roots of Negative Numbers

Expressions Involving i

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