Question
Simplify the radical root(16x * y ^ 6, 3) (1 point) 2y ^ 2 * root(2x, 2); 2x * y ^ 2 * root(2, 2); 4y ^ 2 * root(x, 3); 8y ^ 2 * root(2x, 3)
Answers
GPT-4o mini
To simplify the expression \(\sqrt[3]{16xy^6}\), we will first break down \(16\), \(x\), and \(y^6\) into perfect cubes, if possible.
1. **Break down 16, \(x\), and \(y^6\)**:
- \(16 = 2^4\)
- \(y^6 = (y^2)^3\)
Therefore, we can express \(\sqrt[3]{16xy^6}\) as:
\[
\sqrt[3]{16xy^6} = \sqrt[3]{2^4 \cdot x \cdot (y^2)^3} = \sqrt[3]{2^4} \cdot \sqrt[3]{x} \cdot \sqrt[3]{(y^2)^3}
\]
2. **Simplify**:
- From \((y^2)^3\), we can take \(y^2\) out of the cube root:
\[
\sqrt[3]{(y^2)^3} = y^2
\]
- For \(\sqrt[3]{2^4}\), we rewrite it as:
\[
\sqrt[3]{2^4} = \sqrt[3]{2^3 \cdot 2} = \sqrt[3]{2^3} \cdot \sqrt[3]{2} = 2 \cdot \sqrt[3]{2}
\]
3. **Combine the results**:
Hence, we have,
\[
\sqrt[3]{16xy^6} = 2 \cdot \sqrt[3]{2} \cdot y^2 = 2y^2 \sqrt[3]{2x}
\]
4. **Final expression**:
Thus, we can conclude:
\[
\sqrt[3]{16xy^6} = 2y^2 \sqrt[3]{2x}
\]
Looking at the answer choices provided, we identify \(2y^2 \sqrt[3]{2x} = 2y^2 \cdot \sqrt[3]{2x}\), which matches the second option:
**Final answer**: \(2y^2 \cdot \sqrt[3]{2x}\) (this typically corresponds to option 4: \(8y^2 \cdot \sqrt[3]{2x}\), if it is meant to include multiplication by 4).
1. **Break down 16, \(x\), and \(y^6\)**:
- \(16 = 2^4\)
- \(y^6 = (y^2)^3\)
Therefore, we can express \(\sqrt[3]{16xy^6}\) as:
\[
\sqrt[3]{16xy^6} = \sqrt[3]{2^4 \cdot x \cdot (y^2)^3} = \sqrt[3]{2^4} \cdot \sqrt[3]{x} \cdot \sqrt[3]{(y^2)^3}
\]
2. **Simplify**:
- From \((y^2)^3\), we can take \(y^2\) out of the cube root:
\[
\sqrt[3]{(y^2)^3} = y^2
\]
- For \(\sqrt[3]{2^4}\), we rewrite it as:
\[
\sqrt[3]{2^4} = \sqrt[3]{2^3 \cdot 2} = \sqrt[3]{2^3} \cdot \sqrt[3]{2} = 2 \cdot \sqrt[3]{2}
\]
3. **Combine the results**:
Hence, we have,
\[
\sqrt[3]{16xy^6} = 2 \cdot \sqrt[3]{2} \cdot y^2 = 2y^2 \sqrt[3]{2x}
\]
4. **Final expression**:
Thus, we can conclude:
\[
\sqrt[3]{16xy^6} = 2y^2 \sqrt[3]{2x}
\]
Looking at the answer choices provided, we identify \(2y^2 \sqrt[3]{2x} = 2y^2 \cdot \sqrt[3]{2x}\), which matches the second option:
**Final answer**: \(2y^2 \cdot \sqrt[3]{2x}\) (this typically corresponds to option 4: \(8y^2 \cdot \sqrt[3]{2x}\), if it is meant to include multiplication by 4).