Let us simplify the expression [tex]\(\sqrt{\frac{41 y^5}{16 x^6}}\)[/tex] step-by-step.
1. Identify and separate the square root:
[tex]\[
\sqrt{\frac{41 y^5}{16 x^6}}
\][/tex]
2. Break down the square root into the product of square roots:
[tex]\[
\sqrt{\frac{41 y^5}{16 x^6}} = \sqrt{41} \cdot \sqrt{\frac{y^5}{16 x^6}}
\][/tex]
3. Simplify the square root of the fraction:
[tex]\[
\sqrt{\frac{y^5}{16 x^6}} = \frac{\sqrt{y^5}}{\sqrt{16 x^6}}
\][/tex]
4. Simplify the square roots of the numerator and the denominator separately:
- The square root of [tex]\(16 x^6\)[/tex]:
[tex]\[
\sqrt{16 x^6} = \sqrt{16} \cdot \sqrt{x^6} = 4 \cdot x^3 = 4x^3
\][/tex]
- The square root of [tex]\(y^5\)[/tex]:
[tex]\[
\sqrt{y^5} = \sqrt{y^4 \cdot y} = \sqrt{y^4} \cdot \sqrt{y} = y^2 \cdot \sqrt{y} = y^2 \sqrt{y}
\][/tex]
5. Combine these results:
[tex]\[
\frac{\sqrt{y^5}}{\sqrt{16 x^6}} = \frac{y^2 \sqrt{y}}{4 x^3}
\][/tex]
6. Combine back with [tex]\(\sqrt{41}\)[/tex]:
[tex]\[
\sqrt{41} \cdot \frac{y^2 \sqrt{y}}{4 x^3} = \frac{\sqrt{41} \cdot y^2 \cdot \sqrt{y}}{4 x^3}
\][/tex]
7. Simplify the final expression:
[tex]\[
\frac{\sqrt{41} \cdot y^2 \cdot \sqrt{y}}{4 x^3} = \frac{\sqrt{41} \cdot \sqrt{y^5}}{4 x^3}
\][/tex]
Thus, the simplified expression is:
[tex]\[
\frac{\sqrt{41} \cdot \sqrt{y^5}}{4 x^3}
\][/tex]
