Questions: Identify two expressions that are equivalent to 4-8 i. (A) sqrt(16)+sqrt(-64) (B) -2(sqrt(-16)-sqrt(4)) (C) sqrt(16)-sqrt(-64) (D) -2(sqrt(-16)+sqrt(4))

Transcript text: Identify two expressions that are equivalent to $4-8 i$. (A) $\sqrt{16}+\sqrt{-64}$ (B) $-2(\sqrt{-16}-\sqrt{4})$ (C) $\sqrt{16}-\sqrt{-64}$ (D) $-2(\sqrt{-16}+\sqrt{4})$

Solution

Solution Steps

To determine which expressions are equivalent to $4 - 8i$, we need to simplify each given option and compare the results to $4 - 8i$. We will use the properties of square roots and imaginary numbers ($i = \sqrt{-1}$) to simplify the expressions.

Step 1: Evaluate Each Expression

We start by evaluating the given expressions to see if they are equivalent to $4 - 8i$.

For expression (A): \[ \sqrt{16} + \sqrt{-64} = 4 + 8i \]
For expression (B): \[ -2(\sqrt{-16} - \sqrt{4}) = -2(4i - 2) = -2 \cdot 4i + 4 = 4 - 8i \]
For expression (C): \[ \sqrt{16} - \sqrt{-64} = 4 - 8i \]
For expression (D): \[ -2(\sqrt{-16} + \sqrt{4}) = -2(4i + 2) = -2 \cdot 4i - 4 = -4 - 8i \]

Step 2: Compare with the Target Expression

The target expression is $4 - 8i$. We compare each simplified expression: