Question
\( \sqrt{\frac{10}{11}}=\square \) (Simplify your answer, including any radicals.
Ask by Burgess Rodriguez. in the United States
Jan 22,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\sqrt{\frac{10}{11}} = \frac{\sqrt{110}}{11}
\]
Solution
To simplify the expression \( \sqrt{\frac{10}{11}} \), we can rationalize the denominator. Here's how you can do it step by step:
1. **Separate the Square Root:**
\[
\sqrt{\frac{10}{11}} = \frac{\sqrt{10}}{\sqrt{11}}
\]
2. **Rationalize the Denominator:**
To eliminate the square root in the denominator, multiply both the numerator and the denominator by \( \sqrt{11} \):
\[
\frac{\sqrt{10}}{\sqrt{11}} \times \frac{\sqrt{11}}{\sqrt{11}} = \frac{\sqrt{10} \times \sqrt{11}}{11}
\]
3. **Simplify the Numerator:**
Multiply the radicals in the numerator:
\[
\sqrt{10} \times \sqrt{11} = \sqrt{110}
\]
4. **Final Simplified Form:**
\[
\frac{\sqrt{110}}{11}
\]
So, the simplified form of \( \sqrt{\frac{10}{11}} \) is:
\[
\boxed{\dfrac{\sqrt{110}}{11}}
\]
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Simplify this solution The Deep Dive
The expression \( \sqrt{\frac{10}{11}} \) can be simplified by separating the numerator and the denominator inside the square root. This gives us: \[ \sqrt{\frac{10}{11}} = \frac{\sqrt{10}}{\sqrt{11}} \] Now, if you want to rationalize the denominator, you can multiply the numerator and denominator by \( \sqrt{11} \): \[ \frac{\sqrt{10}}{\sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}} = \frac{\sqrt{110}}{11} \] So, the simplified form of \( \sqrt{\frac{10}{11}} \) is: \[ \frac{\sqrt{10}}{\sqrt{11}} \quad \text{or} \quad \frac{\sqrt{110}}{11} \] Both forms are valid, depending on whether you prefer to keep the denominator as a square root or rationalized.
