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Chapter 3: Problem 9

Express the number in terms of i. $$\sqrt{-98}$$

Short Answer

Expert verified
The expression in terms of \(i\) is \(7i\sqrt{2}\).

Step by step solution

01

Identify the Negative Sign

Recognize that the square root contains a negative number: \(\sqrt{-98}\). Understand that this will involve the imaginary unit \(i\), where \(i = \sqrt{-1}\).
02

Separate the Negative Inside the Radical

Write the expression by separating the negative inside the square root: \(\sqrt{-98} = \sqrt{-1 \cdot 98} \).
03

Apply the Property of Square Roots

Use the property that \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\): \(\sqrt{-1 \cdot 98} = \sqrt{-1} \cdot \sqrt{98}\).
04

Introduce the Imaginary Unit

Replace \sqrt{-1}\ with \i\: \(\sqrt{-1} \cdot \sqrt{98} = i \cdot \sqrt{98}\).
05

Simplify the Remaining Expression

Factorize the 98 to find its square root components that can be simplified: \(\sqrt{98} = \sqrt{49 \cdot 2} = \sqrt{49} \cdot \sqrt{2} = 7 \cdot \sqrt{2}\).
06

Combine All Elements

Combine all elements to form the final expression: \(\sqrt{-98} = i \cdot 7 \cdot \sqrt{2} = 7i\sqrt{2}\).

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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 numbers that include both a real and an imaginary part. The general form of a complex number is written as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit defined by \(i^2 = -1\). In the exercise, the negative sign inside the square root led us to use the imaginary unit \(i\). When we see the square root of a negative number, like \(\sqrt{-98}\), we know it involves complex numbers because of the role of \(i\).
Square Roots
Square roots are mathematical operations used to find a number which, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3 because \(3 \times 3 = 9\). The square root of a negative number involves the imaginary unit \(i\). For example, \(\sqrt{-98}\) is decomposed into \(\sqrt{-1} \cdot \sqrt{98}\). Here, \(\sqrt{-1} = i\). This allows us to transform \(\sqrt{-98}\) into an expression involving \(i\).
Simplifying Radicals
Simplifying radicals is an important skill in algebra. It involves breaking down the square root into its simplest form. For example, \(\sqrt{98}\) can be broken into \(\sqrt{49 \cdot 2}\). Since \(\sqrt{49} = 7\), we can simplify \(\sqrt{98}\) into \(7 \cdot \sqrt{2}\). When dealing with negative numbers, it's crucial to remember to include the imaginary unit \(i\), as shown in the exercise: \(\sqrt{-98} = i \cdot 7 \cdot \sqrt{2} = 7i \sqrt{2}\).

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