38. Efetue as divisões a seguir.

a) [tex]$\left(26 x^{16}+52 x^{13}\right): 13 x^{13}$[/tex], com [tex]$x \neq 0$[/tex].

b) [tex]$\left(-2 a^7+8 a^6\right): 4 a^5$[/tex], com [tex]$a \neq 0$[/tex].



Answer :

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]

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