Question
Simplify each cube root expression. Describe the simplified form of the expression as rational or irrational. In your final answer, include all of your work. 1. ^3√81 2. ^3√-64
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WRITER
Answers
Answered by
GPT-4o mini
To simplify each cube root expression, we'll follow these steps:
1. Identify if the number under the cube root can be expressed as a perfect cube.
2. Simplify the cube root accordingly.
3. 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
1. \(\sqrt[3]{81} = 3\sqrt[3]{3}\) (irrational)
2. \(\sqrt[3]{-64} = -4\) (rational)
1. Identify if the number under the cube root can be expressed as a perfect cube.
2. Simplify the cube root accordingly.
3. 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
1. \(\sqrt[3]{81} = 3\sqrt[3]{3}\) (irrational)
2. \(\sqrt[3]{-64} = -4\) (rational)