Exponent Expressions 607F36

1. **Problem 1:** Simplify and analyze $$x^{-\frac{5}{2}} + \frac{1}{x} + 5$$. 2. **Problem 2:** Simplify $$\left(x^{\frac{1}{2}} - 1\right)^2$$. 3. **Problem 3:** Simplify $$2x^{-\frac{3}{2}} + 5x^{-\frac{1}{2}} + 5$$. --- ### Step 1: Simplify Problem 1 - Expression: $$x^{-\frac{5}{2}} + \frac{1}{x} + 5$$ - Recall that $$x^{-a} = \frac{1}{x^a}$$. Rewrite terms: $$x^{-\frac{5}{2}} = \frac{1}{x^{\frac{5}{2}}}$$ $$\frac{1}{x} = x^{-1}$$ So expression is: $$\frac{1}{x^{\frac{5}{2}}} + \frac{1}{x} + 5$$ No further simplification without common denominator. --- ### Step 2: Simplify Problem 2 - Expression: $$\left(x^{\frac{1}{2}} - 1\right)^2$$ Use formula $$(a-b)^2 = a^2 - 2ab + b^2$$: $$\left(x^{\frac{1}{2}}\right)^2 - 2 \cdot x^{\frac{1}{2}} \cdot 1 + 1^2 = x - 2x^{\frac{1}{2}} + 1$$ --- ### Step 3: Simplify Problem 3 - Expression: $$2x^{-\frac{3}{2}} + 5x^{-\frac{1}{2}} + 5$$ Rewrite negative exponents: $$2 \cdot \frac{1}{x^{\frac{3}{2}}} + 5 \cdot \frac{1}{x^{\frac{1}{2}}} + 5 = \frac{2}{x^{\frac{3}{2}}} + \frac{5}{x^{\frac{1}{2}}} + 5$$ --- ### Final answers: 1. $$\frac{1}{x^{\frac{5}{2}}} + \frac{1}{x} + 5$$ 2. $$x - 2x^{\frac{1}{2}} + 1$$ 3. $$\frac{2}{x^{\frac{3}{2}}} + \frac{5}{x^{\frac{1}{2}}} + 5$$