Question
Simplify. \[ \sqrt{v^{11}} \] Assume that the variable represents a positive real number. \( \square \)
Ask by Newton Osborne. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The simplified form of \(\sqrt{v^{11}}\) is \(v^{5}\sqrt{v}\).
Solution
Calculate or simplify the expression \( \sqrt(v^{11}) \).
Simplify the expression by following steps:
- step0: Solution:
\(\sqrt{v^{11}}\)
- step1: Rewrite the exponent:
\(\sqrt{v^{10+1}}\)
- step2: Expand the expression:
\(\sqrt{v^{10}\times v}\)
- step3: Use the properties of radicals:
\(\sqrt{v^{10}}\times \sqrt{v}\)
- step4: Simplify the root:
\(v^{5}\sqrt{v}\)
The simplified form of \(\sqrt{v^{11}}\) is \(v^{5}\sqrt{v}\).
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Simplify this solution The Deep Dive
To simplify \( \sqrt{v^{11}} \), we can use the property of exponents that states \( \sqrt{x} = x^{1/2} \). Therefore: \[ \sqrt{v^{11}} = v^{11/2} \] Now, we can express \( 11/2 \) as a mixed number: \[ 11/2 = 5 \frac{1}{2} \] This means \( v^{11/2} \) can be written as: \[ v^{11/2} = v^5 \cdot v^{1/2} = v^5 \sqrt{v} \] So, the final simplified form is: \[ \sqrt{v^{11}} = v^5 \sqrt{v} \]
