Question

show the steps to subtract $(2.5 + \sqrt{-36}) - (7 - \sqrt{-4})$. (6 points)
$(2.5 + \boxed{} \cdot \sqrt{36}) - (7 - \boxed{} \cdot \sqrt{4})=
$(2.5 + \boxed{}) - (7 - \boxed{})=
2.5 \boxed{} 7 + 6i \boxed{} 2i = -4.5 + 8i
options: 2i, -, +, $\sqrt{-1}$, 6i, $\sqrt{-1}$

Explanation:

Step1: Recall Imaginary Unit

Recall that \( \sqrt{-a}=\sqrt{-1}\cdot\sqrt{a} \) for \( a>0 \). So for \( \sqrt{-36} \), we have \( \sqrt{-36}=\sqrt{-1}\cdot\sqrt{36} \), and for \( \sqrt{-4}=\sqrt{-1}\cdot\sqrt{4} \).

Step2: Simplify Imaginary Terms

Simplify \( \sqrt{36} = 6 \) and \( \sqrt{4}=2 \). So \( \sqrt{-36}=\sqrt{-1}\cdot6 = 6i \) (since \( i = \sqrt{-1} \)) and \( \sqrt{-4}=\sqrt{-1}\cdot2 = 2i \).

Step3: Substitute and Distribute the Subtraction

The expression \( (2.5+\sqrt{-36})-(7 - \sqrt{-4}) \) becomes \( (2.5 + 6i)-(7 - 2i) \). Distribute the subtraction: \( 2.5+6i - 7 + 2i \).

Step4: Combine Like Terms

Combine the real parts: \( 2.5-7=-4.5 \). Combine the imaginary parts: \( 6i + 2i = 8i \). So the result is \( -4.5 + 8i \).

For the first row of boxes:

• First box (for \( \sqrt{-36} \) coefficient): \( \sqrt{-1} \)
• Second box (for \( \sqrt{-4} \) coefficient): \( \sqrt{-1} \)

For the second row of boxes:

• First box (imaginary part of first complex number): \( 6i \)
• Second box (imaginary part of second complex number): \( 2i \)

For the third row of boxes:

• First operation (between 2.5 and 7): \( - \)
• Second operation (between 6i and 2i): \( + \)

Answer:

First row boxes: \( \sqrt{-1} \), \( \sqrt{-1} \)
Second row boxes: \( 6i \), \( 2i \)
Third row boxes: \( - \), \( + \)
Final result: \( -4.5 + 8i \)