Questions: Write in terms of i. Simplify your answer as much as possible. -√(-54)

Transcript text: ALEKS - Makayla Johnson - Lea Grades fo ๑- www-awy.aleks.com/alekscgi/x/lsl.exe/1o_u-lgNslkasNW8D8A9PV Radicals Using i to rewrite square roots of negative numbers Write in terms of $i$. Simplify your answer as much as possible. \[ -\sqrt{-54} \] $\square$

Solution

Solution Steps

To solve the problem of rewriting the expression $-\sqrt{-54}$ in terms of $i$, we need to recognize that the square root of a negative number can be expressed using the imaginary unit $i$, where $i = \sqrt{-1}$. First, we rewrite $-54$ as $-1 \times 54$. Then, we can express $\sqrt{-54}$ as $\sqrt{-1} \times \sqrt{54}$, which simplifies to $i \times \sqrt{54}$. Finally, we simplify $\sqrt{54}$ by factoring it into $\sqrt{9 \times 6}$, which simplifies further to $3\sqrt{6}$. Therefore, the expression becomes $-3i\sqrt{6}$.

Step 1: Rewrite the Expression

We start with the expression $-\sqrt{-54}$. To rewrite this in terms of $i$, we recognize that $-54$ can be expressed as $-1 \times 54$. Thus, we have: \[ -\sqrt{-54} = -\sqrt{-1 \times 54} = -\sqrt{-1} \times \sqrt{54} = -i \sqrt{54} \]

Step 2: Simplify $\sqrt{54}$

Next, we simplify $\sqrt{54}$. We can factor $54$ as $9 \times 6$: \[ \sqrt{54} = \sqrt{9 \times 6} = \sqrt{9} \times \sqrt{6} = 3\sqrt{6} \] Substituting this back into our expression gives: \[ -\sqrt{-54} = -i \times 3\sqrt{6} = -3i\sqrt{6} \]

Step 3: Evaluate the Expression

The expression $-3i\sqrt{6}$ can be evaluated numerically. The approximate value of $\sqrt{6}$ is $2.4495$, so: \[ -3i\sqrt{6} \approx -3i \times 2.4495 \approx -7.3485i \] This confirms our earlier calculation.