Simplify each expression. All varradies ICpre nonnegative numbers. 23. \( \sqrt[5]{z^{10}} \) 24. \( \sqrt[3]{125 x^{6}} \) 25. \( \sqrt{x_{1}^{8} y^{6}} \) 26. \( \sqrt[3]{m^{6} n^{12}} \)

1. Simplify \( \sqrt[5]{z^{10}} \): \[ \sqrt[5]{z^{10}} = \left(z^{10}\right)^{\frac{1}{5}} = z^{\frac{10}{5}} = z^2. \] 2. Simplify \( \sqrt[3]{125 x^{6}} \): Recognize that \(125 = 5^3\), so \[ \sqrt[3]{125 x^{6}} = \sqrt[3]{5^3 x^6} = 5 \cdot \left(x^6\right)^{\frac{1}{3}} = 5x^{\frac{6}{3}} = 5x^2. \] 3. Simplify \( \sqrt{x_{1}^{8} y^{6}} \): Since the square root corresponds to the exponent \( \frac{1}{2} \), \[ \sqrt{x_{1}^{8} y^6} = \left(x_1^{8}\right)^{\frac{1}{2}} \left(y^6\right)^{\frac{1}{2}} = x_1^{\frac{8}{2}} y^{\frac{6}{2}} = x_1^4 y^3. \] 4. Simplify \( \sqrt[3]{m^{6} n^{12}} \): Taking the cube root gives \[ \sqrt[3]{m^{6} n^{12}} = \left(m^6\right)^{\frac{1}{3}} \left(n^{12}\right)^{\frac{1}{3}} = m^{\frac{6}{3}} n^{\frac{12}{3}} = m^2 n^4. \]