Question

explanation:
in these solution steps, we regroup the negative sign in the radical and rewrite the negative root in terms of i:
\\(\sqrt{64} + \sqrt{-144} = \sqrt{8^2} + \sqrt{(-1) \cdot 12^2}\\) (regroup negative sign)
\\(= 8 + i\sqrt{12^2}\\) (change negative root to i)
\\(= 8 + 12i\\) (simplify)

question 4
part a

enter the correct answer in the box.
rewrite the expression \\(\frac{4 + \sqrt{16 - (4)(5)}}{2}\\) as a complex number in standard form, \\(a + bi\\).

Explanation:

Step1: Calculate the discriminant

First, calculate the value inside the square root: \(16 - (4)(5)=16 - 20=-4\). So the expression becomes \(\frac{4+\sqrt{-4}}{2}\).

Step2: Rewrite the square root of negative number

Rewrite \(\sqrt{-4}\) as \(\sqrt{(-1)\times4}\). Using the property of square roots \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a = - 1\) and \(b = 4\)), we get \(\sqrt{-1}\times\sqrt{4}\). Since \(\sqrt{-1}=i\) and \(\sqrt{4} = 2\), then \(\sqrt{-4}=2i\).

Step3: Substitute and simplify

Substitute \(\sqrt{-4}=2i\) into the expression \(\frac{4+\sqrt{-4}}{2}\), we have \(\frac{4 + 2i}{2}\).
Divide each term in the numerator by 2: \(\frac{4}{2}+\frac{2i}{2}=2 + i\).

Answer:

\(2 + i\)