1. **State the problem:** Simplify the expression $$\frac{-128n^{11} + 36n^{21} - 20n^{16}}{8n^{15}}$$ and express it in the form $$16n^x + Bn^y + Cn$$ where $B$ and $C$ are rational numbers and $x$, $y$ are integers. 2. **Write the expression as separate terms:** $$\frac{-128n^{11}}{8n^{15}} + \frac{36n^{21}}{8n^{15}} - \frac{20n^{16}}{8n^{15}}$$ 3. **Simplify each term by dividing coefficients and subtracting exponents:** - For the first term: $$\frac{-128}{8} \times n^{11-15} = -16n^{-4}$$ - For the second term: $$\frac{36}{8} \times n^{21-15} = \frac{36}{8} n^{6} = \frac{9}{2} n^{6}$$ - For the third term: $$\frac{-20}{8} \times n^{16-15} = -\frac{20}{8} n^{1} = -\frac{5}{2} n$$ 4. **Rewrite the expression:** $$-16n^{-4} + \frac{9}{2} n^{6} - \frac{5}{2} n$$ 5. **Match the expression to the form $$16n^x + Bn^y + Cn$$:** - The term with $16n^x$ corresponds to $-16n^{-4}$, so $x = -4$ and the coefficient is $16$ but with a negative sign, so we can write $16n^x$ as $-16n^{-4}$. - The term with $Bn^y$ corresponds to $\frac{9}{2} n^{6}$, so $B = \frac{9}{2}$ and $y = 6$. - The term with $Cn$ corresponds to $-\frac{5}{2} n$, so $C = -\frac{5}{2}$. **Final answers:** - $x = -4$ - $B = \frac{9}{2}$ - $y = 6$ - $C = -\frac{5}{2}$