1. **Stating the problem:** We are given expressions involving square roots and powers with numbers 15, 6, 0, 43, and powers 2, 3, 4. The goal is to simplify or evaluate the expressions involving these roots and powers. 2. **Understanding the expressions:** The first expression appears to be $15\sqrt{60\times 43} - 45 + 75 + 15 + 0$. The second expression is $15 (60^{4/3})^2$. 3. **Simplify the first expression:** Calculate inside the square root: $$60 \times 43 = 2580$$ So the expression becomes: $$15 \sqrt{2580} - 45 + 75 + 15 + 0$$ 4. **Simplify $\sqrt{2580}$:** Factor 2580: $$2580 = 4 \times 645$$ Since $\sqrt{4} = 2$, we have: $$\sqrt{2580} = \sqrt{4 \times 645} = 2 \sqrt{645}$$ 5. **Substitute back:** $$15 \times 2 \sqrt{645} - 45 + 75 + 15 + 0 = 30 \sqrt{645} + ( -45 + 75 + 15 + 0)$$ Calculate the sum: $$-45 + 75 = 30$$ $$30 + 15 = 45$$ $$45 + 0 = 45$$ So the expression is: $$30 \sqrt{645} + 45$$ 6. **Simplify the second expression:** Given: $$15 (60^{4/3})^2$$ Use the power of a power rule: $$(a^m)^n = a^{m \times n}$$ So: $$ (60^{4/3})^2 = 60^{(4/3) \times 2} = 60^{8/3}$$ 7. **Rewrite the expression:** $$15 \times 60^{8/3}$$ 8. **Express $60^{8/3}$ as a radical:** $$60^{8/3} = (60^{1/3})^8 = (\sqrt[3]{60})^8$$ 9. **Final simplified forms:** - First expression: $$30 \sqrt{645} + 45$$ - Second expression: $$15 \times 60^{8/3}$$ These are the simplified forms based on the given expressions and operations.