Watch the video and then solve the problem given below. Write the number in standard form. $$ \frac{-6+\sqrt{-54}}{3} $$

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UpStudy AI · Accepted Answer

We start with the expression:
$$
\frac{-6+\sqrt{-54}}{3}
$$

**Step 1: Rewrite the square root of a negative number as an imaginary number**

Recognize that:
$$
\sqrt{-54} = \sqrt{54 \cdot (-1)} = \sqrt{54} \cdot \sqrt{-1} = i\sqrt{54}
$$

Next, simplify $$\sqrt{54}$$ by factoring out the largest perfect square, which is $$9$$:
$$
\sqrt{54} = \sqrt{9 \times 6} = \sqrt{9} \cdot \sqrt{6} = 3\sqrt{6}
$$
Thus, we have:
$$
\sqrt{-54} = 3i\sqrt{6}
$$

**Step 2: Substitute back into the expression**

Replace $$\sqrt{-54}$$ with $$3i\sqrt{6}$$:
$$
\frac{-6+3i\sqrt{6}}{3}
$$

**Step 3: Divide each term by $$3$$**

Divide both $$-6$$ and $$3i\sqrt{6}$$ by $$3$$:
$$
\frac{-6}{3} + \frac{3i\sqrt{6}}{3} = -2 + i\sqrt{6}
$$

So, the expression in standard form is:
$$
-2 + i\sqrt{6}
$$
The number in standard form is $$ -2 + i\sqrt{6} $$.

Watch the video and then solve the problem given below. Write the number in standard form. \( \frac{-6+\sqrt{-54}}{3} \)

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