[tex]\left(9x^8y^3\right)^\frac{1}{2}=\left(3^2x^8y^3\right)^\frac{1}{2}=\left(3^2\right)^\frac{1}{2}\cdot\left(x^8\right)^\frac{1}{2}\cdot\left(y^3\right)^\frac{1}{2}\\\\=3^{2\cdot\frac{1}{2}}\cdot x^{8\cdot\frac{1}{2}}\cdot y^{3\cdot\frac{1}{2}}=3^1\cdot x^4\cdot y^\frac{3}{2}=3x^4\cdot y^{1\frac{1}{2}}\\\\=3x^4\cdot y^{1+\frac{1}{2}}=3x^4\cdot y^1\cdot y^\frac{1}{2}=3x^4y\sqrt{y}[/tex]