Sure, let's simplify the given expression step-by-step.
### Simplification of the first expression [tex]\(\left(100 y^9 - 81 y + 32\right) - \left(99 y^9 + 36\right) + 4 y^2 + 15 y^9\)[/tex]:
1. Distribute the negative sign into the second parenthesis:
[tex]\[
100 y^9 - 81 y + 32 - 99 y^9 - 36 + 4 y^2 + 15 y^9
\][/tex]
2. Combine like terms:
- Terms with [tex]\(y^9\)[/tex]: [tex]\(100 y^9 - 99 y^9 + 15 y^9\)[/tex]
- Terms with [tex]\(y^2\)[/tex]: [tex]\(4 y^2\)[/tex]
- Terms with [tex]\(y\)[/tex]: [tex]\(-81 y\)[/tex]
- Constant terms: [tex]\(32 - 36\)[/tex]
3. Simplify each set of like terms:
- For [tex]\(y^9\)[/tex]: [tex]\[ 100 y^9 - 99 y^9 + 15 y^9 = (100 - 99 + 15) y^9 = 16 y^9 \][/tex]
- For [tex]\(y^2\)[/tex]: [tex]\[ 4 y^2 = 4 y^2 \][/tex]
- For [tex]\(y\)[/tex]: [tex]\[ -81 y = -81 y \][/tex]
- For the constants: [tex]\[ 32 - 36 = -4 \][/tex]
4. Combine everything into the simplified expression:
[tex]\[
16 y^9 + 4 y^2 - 81 y - 4
\][/tex]
### Simplification of the second expression [tex]\(\left(\frac{-3 x^2}{11} + \frac{4}{11} \right) + \left(\frac{7 x}{11} - \frac{4}{11} \right) + \left(\frac{3 x^2}{11} - \frac{6 x}{11} \right) + \left(\frac{-24 x}{11} \right) \)[/tex]:
1. Combine like terms:
- Terms with [tex]\(x^2\)[/tex]: [tex]\(\frac{-3 x^2}{11} + \frac{3 x^2}{11}\)[/tex]
- Terms with [tex]\(x\)[/tex]: [tex]\(\frac{7 x}{11} - \frac{6 x}{11} - \frac{24 x}{11}\)[/tex]
- Constant terms: [tex]\(\frac{4}{11} - \frac{4}{11}\)[/tex]
2. Simplify each set of like terms:
- For [tex]\(x^2\)[/tex]: [tex]\[ \frac{-3 x^2}{11} + \frac{3 x^2}{11} = 0 \][/tex]
- For [tex]\(x\)[/tex]: [tex]\[ \frac{7 x}{11} - \frac{6 x}{11} - \frac{24 x}{11} = \frac{7 - 6 - 24}{11} x = \frac{-23 x}{11} \][/tex]
- For the constants: [tex]\[ \frac{4}{11} - \frac{4}{11} = 0 \][/tex]
3. Combine everything into the simplified expression:
[tex]\[
0 + \frac{-23 x}{11} + 0 = -\frac{23 x}{11}
\][/tex]
So, after detailed step-by-step simplification, the answers to the two given expressions are:
1. [tex]\(16 y^9 + 4 y^2 - 81 y - 4\)[/tex]
2. [tex]\(-\frac{23 x}{11}\)[/tex]
