Function Simplification 3Ec937

1. **Problem:** Simplify the function $$h(x) = \frac{3x^4 - 2\sqrt{x} + 5}{x^2}$$. 2. **Formula and rules:** When dividing a polynomial or expression by $$x^2$$, divide each term separately by $$x^2$$. 3. **Step-by-step simplification:** $$h(x) = \frac{3x^4}{x^2} - \frac{2\sqrt{x}}{x^2} + \frac{5}{x^2}$$ 4. Simplify each term: $$\frac{3x^4}{x^2} = 3x^{4-2} = 3x^2$$ $$\frac{2\sqrt{x}}{x^2} = \frac{2x^{\frac{1}{2}}}{x^2} = 2x^{\frac{1}{2} - 2} = 2x^{-\frac{3}{2}}$$ $$\frac{5}{x^2} = 5x^{-2}$$ 5. **Final simplified form:** $$h(x) = 3x^2 - 2x^{-\frac{3}{2}} + 5x^{-2}$$ --- 1. **Problem:** Simplify the function $$m(x) = \left(x - \frac{2}{\sqrt{x}}\right)^2$$. 2. **Formula and rules:** Use the formula for the square of a binomial: $$(a - b)^2 = a^2 - 2ab + b^2$$ 3. **Step-by-step simplification:** Let $$a = x$$ and $$b = \frac{2}{\sqrt{x}} = 2x^{-\frac{1}{2}}$$. Calculate each term: $$a^2 = x^2$$ $$2ab = 2 \times x \times 2x^{-\frac{1}{2}} = 4x^{1 - \frac{1}{2}} = 4x^{\frac{1}{2}}$$ $$b^2 = \left(2x^{-\frac{1}{2}}\right)^2 = 4x^{-1}$$ 4. Substitute back: $$m(x) = x^2 - 4x^{\frac{1}{2}} + 4x^{-1}$$ --- 1. **Problem:** Simplify the function $$p(x) = \sqrt{x} \left(\sqrt[3]{x} - \frac{3}{x^2}\right) + \frac{5}{\sqrt[6]{x}}$$. 2. **Formula and rules:** Express all radicals as exponents: $$\sqrt{x} = x^{\frac{1}{2}}$$ $$\sqrt[3]{x} = x^{\frac{1}{3}}$$ $$\frac{3}{x^2} = 3x^{-2}$$ $$\frac{5}{\sqrt[6]{x}} = 5x^{-\frac{1}{6}}$$ 3. **Step-by-step simplification:** Multiply inside the parentheses: $$x^{\frac{1}{2}} \times x^{\frac{1}{3}} = x^{\frac{1}{2} + \frac{1}{3}} = x^{\frac{3}{6} + \frac{2}{6}} = x^{\frac{5}{6}}$$ $$x^{\frac{1}{2}} \times (-3x^{-2}) = -3x^{\frac{1}{2} - 2} = -3x^{-\frac{3}{2}}$$ 4. Substitute back: $$p(x) = x^{\frac{5}{6}} - 3x^{-\frac{3}{2}} + 5x^{-\frac{1}{6}}$$