Question

9, ^3√200, -√20, order these from least to greatest

Answer 1

To compare the numbers \(9\), \(\sqrt[3]{200}\), and \(-\sqrt{20}\), let's first approximate their values.

1. **Calculating \(9\)**:
- \(9\) is straightforward.

2. **Calculating \(\sqrt[3]{200}\)**:
- We know that \(5^3 = 125\) and \(6^3 = 216\). Therefore, \(\sqrt[3]{200}\) is between \(5\) and \(6\).
- For a better approximation, let's estimate it:
- \(5.8^3 = 195.112\) and \(5.9^3 = 205.379\). Thus, \(\sqrt[3]{200} \approx 5.85\).

3. **Calculating \(-\sqrt{20}\)**:
- We find \(\sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}\).
- Since \(\sqrt{5} \approx 2.236\), we approximate:
- \(-\sqrt{20} \approx -2 \cdot 2.236 \approx -4.472\).

Now we have the approximate values:
- \(9\) (exactly \(9\))
- \(\sqrt[3]{200} \approx 5.85\)
- \(-\sqrt{20} \approx -4.472\)

Now, ordering these from least to greatest:
\[
-\sqrt{20} \approx -4.472 < \sqrt[3]{200} \approx 5.85 < 9
\]

Thus, the order from least to greatest is:
\[
-\sqrt{20}, \sqrt[3]{200}, 9
\]