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

Verified step by step guidance

1. 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.
2. 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 \(i\).
3. Express \(\sqrt{-49}\) as \(\sqrt{49 \times -1}\), which can be separated into \(\sqrt{49} \times \sqrt{-1}\) using the property of square roots.
4. Calculate \(\sqrt{49}\), which is a positive real number, and replace \(\sqrt{-1}\) with \(i\). So, \(\sqrt{-49} = \sqrt{49} \times i\).
5. 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.

Key Concepts

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.