In Exercises 29–36, simplify and write the result in standard form. ___ √−49

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Recognize that the expression involves the square root of a negative number, specifically \(\sqrt{-49}\). This indicates the use of imaginary numbers since the square root of a negative number is not defined in the set of real numbers.

Recall the definition of the imaginary unit \(i\), where \(i = \sqrt{-1}\). This allows us to rewrite the square root of a negative number in terms of \(i\).

Express \(\sqrt{-49}\) as \(\sqrt{49 \times -1}\), which can be separated into \(\sqrt{49} \times \sqrt{-1}\) using the property of square roots.

Calculate \(\sqrt{49}\), which is a positive real number, and replace \(\sqrt{-1}\) with \(i\). So, \(\sqrt{-49} = \sqrt{49} \times i\).

Write the simplified expression in standard form for complex numbers, which is \(a + bi\). Since there is no real part here, the expression will be purely imaginary.

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주요 개념

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Imaginary Numbers

Imaginary numbers extend the real number system by including the square root of negative one, denoted as i. Since the square root of a negative number is not real, it is expressed using i, where i² = -1. For example, √-49 can be written as 7i.

Simplifying Square Roots of Negative Numbers

To simplify the square root of a negative number, separate it into the square root of the negative sign and the square root of the positive number. For instance, √-49 = √-1 × √49 = i × 7 = 7i. This process helps convert complex roots into standard imaginary form.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where a and b are real numbers and i is the imaginary unit. When simplifying expressions like √-49, the result is purely imaginary (0 + 7i). Writing answers in this form clarifies the real and imaginary parts.