Sure, let's solve each division step-by-step.
### Part (a)
We need to simplify the division:
[tex]\[
\frac{26 x^{16} + 52 x^{13}}{13 x^{13}}
\][/tex]
1. Distribute the denominator to both terms in the numerator:
[tex]\[
\frac{26 x^{16}}{13 x^{13}} + \frac{52 x^{13}}{13 x^{13}}
\][/tex]
2. Simplify each fraction:
- For the first term:
[tex]\[
\frac{26 x^{16}}{13 x^{13}} = \frac{26}{13} \cdot \frac{x^{16}}{x^{13}} = 2 \cdot x^{16-13} = 2 x^3
\][/tex]
- For the second term:
[tex]\[
\frac{52 x^{13}}{13 x^{13}} = \frac{52}{13} \cdot \frac{x^{13}}{x^{13}} = 4 \cdot 1 = 4
\][/tex]
3. Combine the simplified terms:
[tex]\[
2 x^3 + 4
\][/tex]
So, the simplified form of [tex]\(\left(26 x^{16} + 52 x^{13}\right) \div 13 x^{13}\)[/tex] is:
[tex]\[
\boxed{2 x^3 + 4}
\][/tex]
### Part (b)
We need to simplify the division:
[tex]\[
\frac{-2 a^7 + 8 a^6}{4 a^5}
\][/tex]
1. Distribute the denominator to both terms in the numerator:
[tex]\[
\frac{-2 a^7}{4 a^5} + \frac{8 a^6}{4 a^5}
\][/tex]
2. Simplify each fraction:
- For the first term:
[tex]\[
\frac{-2 a^7}{4 a^5} = \frac{-2}{4} \cdot \frac{a^7}{a^5} = -\frac{1}{2} \cdot a^{7-5} = -\frac{1}{2} \cdot a^2 = -\frac{1}{2} a^2
\][/tex]
- For the second term:
[tex]\[
\frac{8 a^6}{4 a^5} = \frac{8}{4} \cdot \frac{a^6}{a^5} = 2 \cdot a^{6-5} = 2 \cdot a = 2a
\][/tex]
3. Combine the simplified terms:
[tex]\[
-\frac{1}{2} a^2 + 2a
\][/tex]
So, the simplified form of [tex]\(\left(-2 a^7 + 8 a^6\right) \div 4 a^5\)[/tex] is:
[tex]\[
\boxed{-\frac{1}{2} a^2 + 2a}
\][/tex]
Thus, the answers to the given division problems are:
(a) [tex]\(2 x^3 + 4\)[/tex]
(b) [tex]\(-\frac{1}{2} a^2 + 2a\)[/tex]
