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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].
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].
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