Problem 8 Express each number in terms of ... [FREE SOLUTION]

Chapter 12: Problem 8

Express each number in terms of $j.$ $$-\sqrt{-49}$$

Short Answer

Expert verified

$-\sqrt{-49} = -7j.$

Step by step solution

Identify the square root

We start with the expression $-\sqrt{-49}.$Here, we need to determine the square root of the negative number inside the square root, which is $-49.$ .

Express the negative number using imaginary unit

The imaginary unit is $j$ (or sometimes $i$, but here we'll use $j$).We express $-49$ as $-1 imes 49.$Replacing $-1$ with the imaginary unit $j^2$, gives us:$-49 = j^2 imes 49.$

Simplify the square root

Apply the square root to each component individually:$-\sqrt{j^2 imes 49}\ = -\sqrt{j^2} imes \sqrt{49}.$ Since $\sqrt{j^2} = j$, and $\sqrt{49} = 7$, it becomes:- j \times 7.\.

Multiply to get the final expression

Multiply $j$ by $7$, resulting in:- 7j.Therefore, $-\sqrt{-49} = -7j.$

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Key Concepts

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

Complex Numbers

Complex numbers are an extension of the real number system. They include a real part and an imaginary part, expressed in the form $ a + bj $, where $ a $ and $ b $ are real numbers, and $ j $ is the imaginary unit.

Real Part: The real part of a complex number is the component without the imaginary unit. For example, in $ 3 + 4j $, the real part is 3.
Imaginary Part: The imaginary part, denoted by $ bj $, includes the imaginary unit $ j $. In $ 3 + 4j $, the imaginary part is $ 4j $.

Complex numbers are incredibly useful in many fields, including engineering, physics, and computer science. They allow us to perform arithmetic operations with negative square roots. By extending the number system to include complex numbers, we can solve equations that have no solutions within the realm of real numbers.

Square Roots

The square root of a number $ x $ is a value that, when multiplied by itself, gives $ x $. For example, the square root of 9 is 3 because $ 3 \times 3 = 9 $.When dealing with negative numbers, however, the concept becomes a bit tricky. Negative numbers do not have real square roots because a real number squared is always non-negative. However, by using complex numbers, we extend square roots to negative values using the imaginary unit $ j $. For instance, the expression $ \sqrt{-49} $ can be rewritten as $ \sqrt{j^2 \times 49} $, and it simplifies further because $ \sqrt{j^2} = j $. Thus, the square root of $ -49 $ is expressed as $ 7j $.

Imaginary Unit

The imaginary unit is denoted by $ j $ (or sometimes $ i $ in mathematics and physics). It is defined as the square root of $ -1 $, which can be represented as:\[j = \sqrt{-1}\]Some key properties of the imaginary unit $ j $ include:

Basic Definition: Since $ j^2 = -1 $, it follows that $ j \times j = -1 $.
Arithmetic with $ j $: You can multiply, divide, add, and subtract using $ j $ just like with any algebraic expression. For example, $ 2j + 3j = 5j $.

The imaginary unit is essential for expressing the square roots of negative numbers. For example, in the equation $ \sqrt{-49} $, the imaginary unit transforms a non-real expression into a complex number by allowing us to write it as $ 7j $. This provides a practical way to perform calculations with numbers that were previously considered unsolvable.

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91影视

Express each number in terms of \(j.\) $$-\sqrt{-49}$$

Short Answer

Step by step solution

Identify the square root

Express the negative number using imaginary unit

Simplify the square root

Multiply to get the final expression

Key Concepts

Complex Numbers

Square Roots

Imaginary Unit

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