Question
Simplify. Assume \( x \) is greater than or equal to zero. \( \sqrt{20 x^{3}} \)
Ask by Mcfarlane Harris. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\(2x\sqrt{5x}\)
Solution
To simplify the expression \(\sqrt{20x^{3}}\) with the assumption that \(x \geq 0\), follow these steps:
1. **Factor the Radicand:**
\[
20x^{3} = 4 \times 5 \times x^{2} \times x
\]
Here, \(4\) and \(x^{2}\) are perfect squares.
2. **Apply the Square Root to Each Factor:**
\[
\sqrt{20x^{3}} = \sqrt{4 \times 5 \times x^{2} \times x} = \sqrt{4} \times \sqrt{5} \times \sqrt{x^{2}} \times \sqrt{x}
\]
3. **Simplify Each Square Root:**
\[
\sqrt{4} = 2,\quad \sqrt{x^{2}} = x \quad (\text{since } x \geq 0)
\]
So,
\[
\sqrt{20x^{3}} = 2 \times x \times \sqrt{5x} = 2x\sqrt{5x}
\]
**Final Simplified Form:**
\[
2x\sqrt{5x}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To simplify \( \sqrt{20 x^{3}} \), we start by breaking it down. We can write \( 20 \) as \( 4 \times 5 \), so: \[ \sqrt{20 x^{3}} = \sqrt{4 \times 5 \times x^{3}} \] Next, we can separate the square root of each term: \[ \sqrt{20 x^{3}} = \sqrt{4} \times \sqrt{5} \times \sqrt{x^{3}} \] Now, we know that \( \sqrt{4} = 2 \) and \( \sqrt{x^{3}} = \sqrt{x^{2} \cdot x} = x \sqrt{x} \). Therefore, we can rewrite the expression as: \[ \sqrt{20 x^{3}} = 2 \times \sqrt{5} \times x \sqrt{x} = 2x \sqrt{5x} \] So, the simplified form of \( \sqrt{20 x^{3}} \) is: \[ 2x \sqrt{5x} \]
