Question
Simplify. Assume \( q \) is greater than or equal to zero. \( \sqrt{75 q^{3}} \)
Ask by Robbins Bartlett. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( 5q\sqrt{3q} \)
Solution
To simplify \( \sqrt{75 q^{3}} \) with the assumption that \( q \geq 0 \), follow these steps:
1. **Factor the radicand (the expression inside the square root)**:
\[
75 q^3 = 25 \times 3 \times q^2 \times q
\]
2. **Separate the square root using the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \)**:
\[
\sqrt{75 q^3} = \sqrt{25 \times 3 \times q^2 \times q} = \sqrt{25} \times \sqrt{q^2} \times \sqrt{3q}
\]
3. **Simplify each square root**:
\[
\sqrt{25} = 5, \quad \sqrt{q^2} = q, \quad \text{and} \quad \sqrt{3q} \text{ remains as is since it cannot be simplified further.}
\]
4. **Combine the simplified terms**:
\[
5 \times q \times \sqrt{3q} = 5q\sqrt{3q}
\]
**Final Simplified Form**:
\[
5q\sqrt{3q}
\]
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Simplify this solution Extra Insights
To simplify \( \sqrt{75 q^{3}} \), we start by breaking down the expression inside the square root. Notice that \( 75 = 25 \times 3 \) and \( q^{3} = q^{2} \times q \). So, we rewrite the expression: \[ \sqrt{75 q^{3}} = \sqrt{25 \times 3 \times q^{2} \times q} \] Now, we can separate the square root into parts: \[ = \sqrt{25} \times \sqrt{3} \times \sqrt{q^{2}} \times \sqrt{q} \] Calculating the square roots of the perfect squares: \[ = 5 \times \sqrt{3} \times q \times \sqrt{q} \] Combining those together: \[ = 5q\sqrt{3q} \] Thus, the simplified expression is: \[ 5q\sqrt{3q} \]
