To simplify the expression \(\sqrt[3]{-72}\), we can start by factoring \(-72\).
First, we can rewrite \(-72\) as:
\[ -72 = -1 \times 72 \]
Next, we can further factor \(72\):
\[ 72 = 8 \times 9 = 8 \times (3^2) \]
Since \(8\) is a perfect cube (\(2^3\)), we can express \(-72\) as:
\[ -72 = -1 \times 2^3 \times 3^2 \]
Now we can use the property of cube roots that states:
\[ \sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b} \]
Applying this to our expression, we have:
\[ \sqrt[3]{-72} = \sqrt[3]{-1} \times \sqrt[3]{2^3} \times \sqrt[3]{3^2} \]
Calculating each component:
- \(\sqrt[3]{-1} = -1\)
- \(\sqrt[3]{2^3} = 2\)
- \(\sqrt[3]{3^2} = 3^{2/3} = 3^{2/3} = \sqrt[3]{9}\)
Now we can combine these results:
\[ \sqrt[3]{-72} = -1 \times 2 \times \sqrt[3]{9} = -2 \sqrt[3]{9} \]
So, the simplified form of \(\sqrt[3]{-72}\) is:
\[ \sqrt[3]{-72} = -2 \sqrt[3]{9} \]