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

**23. Simplify \( \sqrt[5]{z^{10}} \)** - Express the radical in exponential form: \[ \sqrt[5]{z^{10}} = z^{\frac{10}{5}} \] - Simplify the exponent: \[ z^{\frac{10}{5}} = z^2 \] - **Answer:** \( z^2 \) --- **24. Simplify \( \sqrt[3]{125 x^{6}} \)** - Factor the radicand into perfect cubes: \[ 125 = 5^3 \quad \text{and} \quad x^{6} = \left(x^2\right)^3 \] - Write the whole expression as: \[ \sqrt[3]{125 x^6} = \sqrt[3]{5^3\,(x^2)^3} \] - Apply the cube root to each factor: \[ \sqrt[3]{5^3} = 5 \quad \text{and} \quad \sqrt[3]{(x^2)^3} = x^2 \] - Multiply the results: \[ 5 \cdot x^2 = 5x^2 \] - **Answer:** \( 5x^2 \) --- **25. Simplify \( \sqrt{x^{8} y^{6}} \)** - Express each variable's exponent under the square root as a fractional exponent: \[ \sqrt{x^{8} y^{6}} = x^{\frac{8}{2}}y^{\frac{6}{2}} \] - Simplify the exponents: \[ x^{\frac{8}{2}} = x^4 \quad \text{and} \quad y^{\frac{6}{2}} = y^3 \] - Multiply the results: \[ x^4 y^3 \] - **Answer:** \( x^4y^3 \) --- **26. Simplify \( \sqrt[3]{m^{6} n^{12}} \)** - Write each term with a fractional exponent by applying the cube root: \[ \sqrt[3]{m^{6}} = m^{\frac{6}{3}} \quad \text{and} \quad \sqrt[3]{n^{12}} = n^{\frac{12}{3}} \] - Simplify the exponents: \[ m^{\frac{6}{3}} = m^2 \quad \text{and} \quad n^{\frac{12}{3}} = n^4 \] - Multiply the results: \[ m^2 n^4 \] - **Answer:** \( m^2n^4 \)