College

13. Which of the following is the correct notation for the complex number [tex]76+\sqrt{-49}[/tex]?

A. [tex]76+i \sqrt{49}[/tex]

B. [tex]76+7i[/tex]

C. [tex]7i+76[/tex]

D. [tex]76-7i[/tex]

Answer :

To solve this problem, let's break down the expression [tex]\(76 + \sqrt{-49}\)[/tex].

1. Understand the square root of a negative number:
- The expression [tex]\(\sqrt{-49}\)[/tex] involves taking the square root of a negative number. In mathematics, the square root of a negative number involves the imaginary unit [tex]\(i\)[/tex], where [tex]\(i = \sqrt{-1}\)[/tex].

2. Simplify the square root:
- First, simplify the positive part: [tex]\(\sqrt{49} = 7\)[/tex].
- Therefore, [tex]\(\sqrt{-49} = \sqrt{49} \cdot \sqrt{-1} = 7i\)[/tex].

3. Combine with the real part:
- Now, substitute [tex]\(7i\)[/tex] back into the original expression: [tex]\(76 + \sqrt{-49}\)[/tex] becomes [tex]\(76 + 7i\)[/tex].

The expression is now written in the standard form of a complex number, [tex]\(a + bi\)[/tex], where [tex]\(a\)[/tex] is the real part and [tex]\(bi\)[/tex] is the imaginary part.

Therefore, the correct notation for the complex number is [tex]\(76 + 7i\)[/tex]. The second and third options, [tex]\(76+7i\)[/tex] and [tex]\(7i+76\)[/tex], denote the same complex number because addition is commutative, but traditionally, we write the real part first. So, the most widely accepted form is [tex]\(76 + 7i\)[/tex].