Question

what is \\(\sqrt{-16} + \sqrt{49}\\) written as a complex number in the form \\(a + bi\\)?
\\(\bigcirc\\) a. \\(7 + 4i\\)
\\(\bigcirc\\) b. \\(-4 + 7i\\)
\\(\bigcirc\\) c. \\(4 + 7i\\)
\\(\bigcirc\\) d. \\(7 - 4i\\)

Explanation:

Step1: Simplify \(\sqrt{-16}\)

Recall that for a negative number under the square root, we can use the imaginary unit \(i = \sqrt{-1}\). So, \(\sqrt{-16}=\sqrt{16\times(-1)}=\sqrt{16}\times\sqrt{-1}\). Since \(\sqrt{16} = 4\) and \(\sqrt{-1}=i\), we have \(\sqrt{-16}=4i\).

Step2: Simplify \(\sqrt{49}\)

The square root of 49 is a real number. We know that \(7\times7 = 49\), so \(\sqrt{49}=7\).

Step3: Add the two results

Now we add \(\sqrt{-16}\) and \(\sqrt{49}\) together. From Step 1, \(\sqrt{-16}=4i\) and from Step 2, \(\sqrt{49}=7\). So, \(\sqrt{-16}+\sqrt{49}=7 + 4i\)? Wait, no, wait. Wait, \(\sqrt{49}\) is 7 and \(\sqrt{-16}\) is \(4i\), so when we add them, it's \(7+4i\)? Wait, no, let's check again. Wait, the expression is \(\sqrt{-16}+\sqrt{49}\), so that's \(4i + 7\), which is \(7 + 4i\) (since addition is commutative). Wait, but let's check the options. Option A is \(7 + 4i\), but wait, maybe I made a mistake. Wait, no, \(\sqrt{-16}=4i\) and \(\sqrt{49}=7\), so adding them gives \(7 + 4i\), which is option A? Wait, but let's check the options again. Wait, the options are:

A. \(7 + 4i\)

B. \(-4 + 7i\)

C. \(4 + 7i\)

D. \(7 - 4i\)

Wait, maybe I mixed up the order? Wait, the expression is \(\sqrt{-16}+\sqrt{49}\), so that's \(4i + 7\), which is \(7 + 4i\), so option A. But wait, let's re - check the simplification of \(\sqrt{-16}\). \(\sqrt{-16}=\sqrt{(-1)\times16}=\sqrt{-1}\times\sqrt{16}=i\times4 = 4i\). And \(\sqrt{49}=7\). So adding them: \(4i+7 = 7 + 4i\), which is option A. Wait, but maybe I misread the expression? The problem says "what is \(\sqrt{-16}+\sqrt{49}\) written as a complex number in the form \(a + bi\)". So \(a\) is the real part and \(b\) is the coefficient of \(i\). So the real part is 7 (from \(\sqrt{49}\)) and the imaginary part is 4 (from \(\sqrt{-16}=4i\)), so the complex number is \(7+4i\), which is option A. Wait, but let's check again. Wait, maybe I had the order wrong. The expression is \(\sqrt{-16}+\sqrt{49}\), so that's \(4i + 7\), which is \(7 + 4i\), so option A.

Answer:

A. \(7 + 4i\)