Question
Simplify the following radical expression. \( \sqrt[3]{125 x^{3} y} \) a.) \( 5 x^{3} y \) b.) \( 5 x_{\sqrt[3]{y}} \) c.) \( 5 x y^{3} \) d. \( 5 \sqrt[3]{x y} \) c.
Ask by Luna Hodgson. in the United States
Jan 26,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The simplified form of the expression is \( 5x \sqrt[3]{y} \).
Solution
Let's simplify the given radical expression step by step:
\[ \sqrt[3]{125 x^{3} y} \]
1. **Factor the Radicand:**
- \(125 = 5^3\)
- \(x^3\) is already a perfect cube.
- \(y\) is not a perfect cube.
2. **Rewrite the Expression Using Exponents:**
\[
\sqrt[3]{5^3 \cdot x^3 \cdot y} = \sqrt[3]{5^3} \cdot \sqrt[3]{x^3} \cdot \sqrt[3]{y}
\]
3. **Simplify Each Component:**
- \(\sqrt[3]{5^3} = 5\)
- \(\sqrt[3]{x^3} = x\)
- \(\sqrt[3]{y}\) remains as is because \(y\) is not a perfect cube.
4. **Combine the Simplified Terms:**
\[
5 \cdot x \cdot \sqrt[3]{y} = 5x \sqrt[3]{y}
\]
**Therefore, the simplified form of the expression is:**
**b.) \( 5x \sqrt[3]{y} \)**
It appears there might have been a misunderstanding in selecting the correct answer. Option **c.) \( 5x y^{3} \)** is incorrect because it incorrectly elevates \( y \) to the power of 3 outside the radical, rather than keeping it inside the cube root.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To simplify \( \sqrt[3]{125 x^{3} y} \), we start by breaking it down. The cube root of 125 is 5, and since \( x^3 \) is also a perfect cube, its cube root is simply \( x \). Thus, we have \( \sqrt[3]{125} = 5 \) and \( \sqrt[3]{x^3} = x \). This leaves us with \( \sqrt[3]{y} \). When combined, this results in: \[ 5x\sqrt[3]{y} \] So the simplified expression is \( 5 x \sqrt[3]{y} \), which corresponds to option b.) \( 5 x_{\sqrt[3]{y}} \).
