Sure, let's simplify the expression [tex]\(y = \left(\frac{21}{x^6} - 3 x^{11}\right)\)[/tex] step-by-step:
1. Start with the given expression:
[tex]\[
y = \frac{21}{x^6} - 3 x^{11}
\][/tex]
2. Combine the terms using a common denominator:
We notice that both terms can be written with a common denominator of [tex]\(x^6\)[/tex]. This means we can rewrite the expression as a single fraction.
[tex]\[
y = \frac{21}{x^6} - \frac{3 x^{11} \cdot x^6}{x^6}
\][/tex]
3. Simplify the second term in the numerator:
We multiply [tex]\(3 x^{11}\)[/tex] by [tex]\(x^6\)[/tex] in the numerator:
[tex]\[
3 x^{11} \cdot x^6 = 3 x^{17}
\][/tex]
So the expression becomes:
[tex]\[
y = \frac{21}{x^6} - \frac{3 x^{17}}{x^6}
\][/tex]
4. Combine under a single denominator:
[tex]\[
y = \frac{21 - 3 x^{17}}{x^6}
\][/tex]
5. Factor out any common terms in the numerator:
The numerator [tex]\(21 - 3 x^{17}\)[/tex] can be factored:
[tex]\[
21 - 3 x^{17} = 3(7 - x^{17})
\][/tex]
Now the expression is:
[tex]\[
y = \frac{3 (7 - x^{17})}{x^6}
\][/tex]
6. Write the final simplified expression:
[tex]\[
y = \frac{3 (7 - x^{17})}{x^6}
\][/tex]
So, the simplified form of [tex]\( \left(\frac{21}{x^6} - 3 x^{11}\right) \)[/tex] is:
[tex]\[
y = \frac{3 (7 - x^{17})}{x^6}
\][/tex]
