1. **State the problem:** Simplify the expression $$6x^{\frac{1}{2}} - 9x^{-\frac{1}{2}} + 15x^{-\frac{3}{2}}$$. 2. **Recall the rules:** - $x^{a}$ means $x$ raised to the power $a$. - Negative exponents mean reciprocal powers: $x^{-a} = \frac{1}{x^a}$. - We can factor expressions by taking out common factors. 3. **Identify the common factor:** The smallest exponent among $\frac{1}{2}, -\frac{1}{2}, -\frac{3}{2}$ is $-\frac{3}{2}$. 4. **Factor out $x^{-\frac{3}{2}}$:** $$6x^{\frac{1}{2}} - 9x^{-\frac{1}{2}} + 15x^{-\frac{3}{2}} = x^{-\frac{3}{2}}\left(6x^{\frac{1}{2} + \frac{3}{2}} - 9x^{-\frac{1}{2} + \frac{3}{2}} + 15\right)$$ 5. **Simplify exponents inside parentheses:** - $\frac{1}{2} + \frac{3}{2} = 2$ - $-\frac{1}{2} + \frac{3}{2} = 1$ So, $$= x^{-\frac{3}{2}}(6x^{2} - 9x + 15)$$ 6. **Factor the polynomial inside parentheses if possible:** $$6x^{2} - 9x + 15 = 3(2x^{2} - 3x + 5)$$ 7. **Final simplified form:** $$= 3x^{-\frac{3}{2}}(2x^{2} - 3x + 5)$$ This is the simplified expression with the common factor taken out. **Answer:** $$3x^{-\frac{3}{2}}(2x^{2} - 3x + 5)$$