Question

Find the cube root -729:3sqrt-729

Answer 1

To find the cube root of -729 and then divide it by \(3\sqrt{-729}\), we first compute each part separately.

1. **Finding the cube root of -729:**
\[
\sqrt[3]{-729} = -9
\]
This is because \((-9) \times (-9) \times (-9) = -729\).

2. **Finding \(3\sqrt{-729}\):**
First, we calculate \(\sqrt{-729}\):
\[
\sqrt{-729} = \sqrt{729} \cdot \sqrt{-1} = 27i
\]
where \(i\) is the imaginary unit.

Now, multiplying by 3:
\[
3\sqrt{-729} = 3 \cdot 27i = 81i
\]

3. **Now, divide the cube root by \(3\sqrt{-729}\):**
\[
\frac{\sqrt[3]{-729}}{3\sqrt{-729}} = \frac{-9}{81i} = \frac{-1}{9i}
\]

4. **To simplify \(\frac{-1}{9i}\), multiply the numerator and denominator by \(-i\) to eliminate the imaginary unit from the denominator:**
\[
\frac{-1 \cdot (-i)}{9i \cdot (-i)} = \frac{i}{-9i^2}
\]
Since \(i^2 = -1\), this simplifies to:
\[
= \frac{i}{9}
\]

Thus, the final result is:
\[
\frac{i}{9}
\]