1. **State the problems:** - Problem 28: Simplify $$\frac{(2x^{9})^{3}(x^{2}y)^{-2}}{x^{3}y^{0}}$$ - Problem 29: Simplify $$\frac{(4x^{-2})^{2}(x^{2}y^{0})^{-3}}{x^{-2}y^{-2.14}}$$ - Problem 30: Evaluate $$(1-3 - 4^{0}) - (-3 - 2) - \sqrt{25}$$ 2. **Recall rules and formulas:** - Power of a power: $$(a^{m})^{n} = a^{mn}$$ - Negative exponents: $$a^{-m} = \frac{1}{a^{m}}$$ - Zero exponent: $$a^{0} = 1$$ - Multiplying powers with same base: $$a^{m} \cdot a^{n} = a^{m+n}$$ - Division of powers with same base: $$\frac{a^{m}}{a^{n}} = a^{m-n}$$ - Square root: $$\sqrt{a} = a^{\frac{1}{2}}$$ --- ### Problem 28 3. Simplify numerator: $$(2x^{9})^{3} = 2^{3} x^{9 \times 3} = 8x^{27}$$ $$(x^{2}y)^{-2} = x^{2 \times (-2)} y^{-2} = x^{-4} y^{-2}$$ 4. Multiply numerator terms: $$8x^{27} \cdot x^{-4} y^{-2} = 8x^{27-4} y^{-2} = 8x^{23} y^{-2}$$ 5. Denominator: $$x^{3} y^{0} = x^{3} \cdot 1 = x^{3}$$ 6. Divide numerator by denominator: $$\frac{8x^{23} y^{-2}}{x^{3}} = 8x^{23-3} y^{-2} = 8x^{20} y^{-2}$$ **Final answer for 28:** $$8x^{20} y^{-2}$$ --- ### Problem 29 3. Simplify numerator: $$(4x^{-2})^{2} = 4^{2} x^{-2 \times 2} = 16 x^{-4}$$ $$(x^{2} y^{0})^{-3} = x^{2 \times (-3)} y^{0 \times (-3)} = x^{-6} y^{0} = x^{-6}$$ 4. Multiply numerator terms: $$16 x^{-4} \cdot x^{-6} = 16 x^{-4-6} = 16 x^{-10}$$ 5. Denominator: $$x^{-2} y^{-2.14}$$ 6. Divide numerator by denominator: $$\frac{16 x^{-10}}{x^{-2} y^{-2.14}} = 16 x^{-10 - (-2)} y^{0 - (-2.14)} = 16 x^{-8} y^{2.14}$$ **Final answer for 29:** $$16 x^{-8} y^{2.14}$$ --- ### Problem 30 3. Evaluate each term: $$4^{0} = 1$$ $$\sqrt{25} = 5$$ 4. Substitute and simplify step-by-step: $$(1 - 3 - 1) - (-3 - 2) - 5$$ Calculate inside parentheses: $$1 - 3 - 1 = (1 - 3) - 1 = -2 - 1 = -3$$ $$-3 - 2 = -5$$ 5. Substitute back: $$-3 - (-5) - 5 = -3 + 5 - 5 = 2 - 5 = -3$$ **Final answer for 30:** $$-3$$