Question

question 9
simplify.
$4i\sqrt{-24}$
$\circ -8i\sqrt{6}$
$\circ 8\sqrt{6}$
$\circ 8i\sqrt{6}$
$\circ -8\sqrt{6}$

Explanation:

Step1: Simplify the square root of a negative number

Recall that \(\sqrt{-a} = i\sqrt{a}\) for \(a>0\). So, \(\sqrt{-24}=\sqrt{24}\times i\). Now, simplify \(\sqrt{24}\): \(\sqrt{24}=\sqrt{4\times6} = \sqrt{4}\times\sqrt{6}=2\sqrt{6}\). Thus, \(\sqrt{-24}=i\times2\sqrt{6} = 2i\sqrt{6}\).

Step2: Multiply with the coefficient

We have the expression \(4i\sqrt{-24}\). Substitute \(\sqrt{-24}=2i\sqrt{6}\) into it: \(4i\times(2i\sqrt{6})\). Multiply the coefficients and the \(i\) terms: \(4\times2 = 8\) and \(i\times i=i^{2}\). Recall that \(i^{2}=-1\). So, \(8\times i^{2}\sqrt{6}=8\times(-1)\sqrt{6}=-8\sqrt{6}\)? Wait, no, wait. Wait, step1 was wrong. Wait, \(\sqrt{-24}=\sqrt{24}\times i\), but \(24 = 4\times6\), so \(\sqrt{24}=2\sqrt{6}\), so \(\sqrt{-24}=i\times2\sqrt{6}\). Then \(4i\times\sqrt{-24}=4i\times(2i\sqrt{6})\). Wait, no, original expression is \(4i\sqrt{-24}\), so it's \(4i\times\sqrt{-24}\). So \(\sqrt{-24}=i\sqrt{24}=i\times2\sqrt{6}=2i\sqrt{6}\). Then \(4i\times2i\sqrt{6}=8i^{2}\sqrt{6}\). Since \(i^{2}=-1\), this is \(8\times(-1)\sqrt{6}=-8\sqrt{6}\)? But that's one of the options? Wait, no, wait, maybe I messed up the sign. Wait, no, let's start over.

Wait, the correct way: \(\sqrt{-24}=\sqrt{24}\times\sqrt{-1}=2\sqrt{6}\times i\). Then \(4i\times\sqrt{-24}=4i\times2i\sqrt{6}=8i^{2}\sqrt{6}\). Since \(i^{2}=-1\), so \(8\times(-1)\sqrt{6}=-8\sqrt{6}\)? But option D is \(-8\sqrt{6}\)? Wait, but let's check again. Wait, maybe I made a mistake in the sign. Wait, no, let's compute again.

Wait, \(4i\sqrt{-24}\). Let's rewrite \(\sqrt{-24}\) as \(\sqrt{24}\times\sqrt{-1}=2\sqrt{6}i\). Then multiply by \(4i\): \(4i\times2\sqrt{6}i = 8\sqrt{6}i^{2}\). Since \(i^{2}=-1\), this is \(8\sqrt{6}\times(-1)=-8\sqrt{6}\). So the answer is \(-8\sqrt{6}\), which is the last option. Wait, but let's check the options. The options are:

• \(-8i\sqrt{6}\)

• \(8\sqrt{6}\)

• \(8i\sqrt{6}\)

• \(-8\sqrt{6}\)

So the correct answer is \(-8\sqrt{6}\), which is the fourth option. Wait, but let's confirm.

Wait, another way: \(4i\sqrt{-24}=4i\times\sqrt{24}\times i=4i^{2}\sqrt{24}\). Since \(i^{2}=-1\), this is \(4\times(-1)\times2\sqrt{6}=-8\sqrt{6}\). Yes, that's correct. So the answer is \(-8\sqrt{6}\), which is the option D (the last one).

Wait, but let's check the options again. The options are:

1. \(-8i\sqrt{6}\)

1. \(8\sqrt{6}\)

1. \(8i\sqrt{6}\)

1. \(-8\sqrt{6}\)

So the correct answer is the fourth option, \(-8\sqrt{6}\).

Answer:

D. \(-8\sqrt{6}\)