Question

Find the cube root of −729 : −729−−−−√3

Answer 1

are you sure

Answer 2

To find the cube root of \(-729\), you are looking for a number \(x\) such that \(x^3 = -729\).

Consider the equation:

\[ x^3 = -729 \]

First, realize that since \(729\) is a positive number and you need \(-729\), the cube root must be a negative number. You can find the positive cube root of \(729\) first and then apply the negative sign.

\[ 729 = 9 \times 9 \times 9 \]

This means:

\[ 9^3 = 729 \]

So:

\[ \sqrt[3]{729} = 9 \]

Since we know that \(x^3\) must equal to \(-729\), we take the negative of the positive cube root. Thus:

\[ \sqrt[3]{-729} = -9 \]

So, the cube root of \(-729\) is:

\[ \boxed{-9} \]

Answer 3

Yes, I'm sure. To verify, let’s follow through the calculations methodically once more:

1. **Identify the cube root equation:**
We are looking for a number \( x \) such that \( x^3 = -729 \).

2. **Consider the absolute value:**
The absolute value of the number is \( 729 \).

3. **Find the cube root of the positive value \( 729 \):**
\[ 9 \times 9 \times 9 = 729 \]
Therefore, \(\sqrt[3]{729} = 9\).

4. **Apply the sign:**
Because we need the cube root of \(-729\) instead of \(729\), and the cube root of a negative number is also negative:
\[ \sqrt[3]{-729} = -9 \]

5. **Verify:**
Calculate \((-9)^3\):
\[ (-9) \times (-9) \times (-9) = -729 \]

Since \((-9)^3 = -729\), the cube root of \(-729\) is indeed \(\boxed{-9}\).