To simplify each cube root expression, we'll follow these steps:
- Identify if the number under the cube root can be expressed as a perfect cube.
- Simplify the cube root accordingly.
- Determine whether the simplified form is rational or irrational.
1. Simplifying \(\sqrt[3]{81}\)
Step 1: Factor \(81\)
We can express \(81\) using its prime factors:
\[ 81 = 3^4 \]
Step 2: Rewrite the cube root
The cube root of \(81\) can be expressed as:
\[ \sqrt[3]{81} = \sqrt[3]{3^4} \]
Step 3: Simplify using properties of exponents
We know that \(3^4\) can be split into a perfect cube and a remaining factor:
\[ 3^4 = (3^3) \cdot (3^1) \]
Thus, we can break it down:
\[ \sqrt[3]{3^4} = \sqrt[3]{3^3 \cdot 3^1} = \sqrt[3]{3^3} \cdot \sqrt[3]{3^1} \]
Calculating \(\sqrt[3]{3^3}\):
\[ \sqrt[3]{3^3} = 3 \]
So we now have:
\[ \sqrt[3]{81} = 3 \cdot \sqrt[3]{3} \text{ (since the remaining part is } 3^1) \]
Conclusion for \(\sqrt[3]{81}\)
\[ \sqrt[3]{81} = 3\sqrt[3]{3} \]
The term \(3\) is rational, but \(\sqrt[3]{3}\) is irrational because \(3\) is not a perfect cube. Therefore, the entire expression \(3\sqrt[3]{3}\) is irrational.
2. Simplifying \(\sqrt[3]{-64}\)
Step 1: Factor \(-64\)
We can express \(-64\) using its prime factors:
\[ -64 = -1 \cdot 4^3 = -1 \cdot (2^3)^3 \]
Step 2: Rewrite the cube root
\[ \sqrt[3]{-64} = \sqrt[3]{-1 \cdot (2^3)^3} = \sqrt[3]{-1} \cdot \sqrt[3]{64} \]
Step 3: Simplify using properties of exponents
We know:
\[ \sqrt[3]{64} = \sqrt[3]{4^3} = 4 \]
Thus, we now have:
\[ \sqrt[3]{-64} = \sqrt[3]{-1} \cdot 4 \]
Calculating \(\sqrt[3]{-1}\):
\[ \sqrt[3]{-1} = -1 \]
Conclusion for \(\sqrt[3]{-64}\)
\[ \sqrt[3]{-64} = -1 \cdot 4 = -4 \]
The value \(-4\) is rational.
Final Answers
- \(\sqrt[3]{81} = 3\sqrt[3]{3}\) (irrational)
- \(\sqrt[3]{-64} = -4\) (rational)