Question

1. \\(\dfrac{42n^7 + 48n^6 + 18n^5}{6n^2}\\)
2. \\(\dfrac{12n^{11} + 48n^9 + 36n^3}{6n^7}\\)
3. \\(\dfrac{24a^8 + 48a^7 + 42a^6}{6a^{12}}\\)
4. \\(\dfrac{20x^9 + 32x^4 + 28x^3}{4x^5}\\)
5. \\(\dfrac{6v^4 + 18v^3 + 36v^2}{6v^2}\\)
6. \\(\dfrac{81n^4 + 3n^3 + 9n^2}{9n^8}\\)

Explanation:

Problem 5

Step1: Factor numerator GCF

$42n^7 + 48n^6 + 18n^5 = 6n^5(7n^2 + 8n + 3)$

Step2: Divide by denominator

$\frac{6n^5(7n^2 + 8n + 3)}{6n^2} = n^{5-2}(7n^2 + 8n + 3)$

Step3: Simplify exponent

$n^3(7n^2 + 8n + 3) = 7n^5 + 8n^4 + 3n^3$

Problem 6

Step1: Factor numerator GCF

$12n^{11} + 48n^9 + 36n^3 = 12n^3(n^8 + 4n^6 + 3)$

Step2: Divide by denominator

$\frac{12n^3(n^8 + 4n^6 + 3)}{6n^7} = 2n^{3-7}(n^8 + 4n^6 + 3)$

Step3: Simplify exponent

$2n^{-4}(n^8 + 4n^6 + 3) = \frac{2n^8 + 8n^6 + 6}{n^4} = 2n^4 + 8n^2 + \frac{6}{n^4}$

Problem 7

Step1: Factor numerator GCF

$24a^8 + 48a^7 + 42a^6 = 6a^6(4a^2 + 8a + 7)$

Step2: Divide by denominator

$\frac{6a^6(4a^2 + 8a + 7)}{6a^{12}} = a^{6-12}(4a^2 + 8a + 7)$

Step3: Simplify exponent

$a^{-6}(4a^2 + 8a + 7) = \frac{4a^2 + 8a + 7}{a^6} = \frac{4}{a^4} + \frac{8}{a^5} + \frac{7}{a^6}$

Problem 8

Step1: Factor numerator GCF

$20x^9 + 32x^4 + 28x^3 = 4x^3(5x^6 + 8x + 7)$

Step2: Divide by denominator

$\frac{4x^3(5x^6 + 8x + 7)}{4x^5} = x^{3-5}(5x^6 + 8x + 7)$

Step3: Simplify exponent

$x^{-2}(5x^6 + 8x + 7) = \frac{5x^6 + 8x + 7}{x^2} = 5x^4 + \frac{8}{x} + \frac{7}{x^2}$

Problem 9

Step1: Factor numerator GCF

$6v^4 + 18v^3 + 36v^2 = 6v^2(v^2 + 3v + 6)$

Step2: Divide by denominator

$\frac{6v^2(v^2 + 3v + 6)}{6v^2} = v^2 + 3v + 6$

Problem 10

Step1: Factor numerator GCF

$81n^4 + 3n^3 + 9n^2 = 3n^2(27n^2 + n + 3)$

Step2: Divide by denominator

$\frac{3n^2(27n^2 + n + 3)}{9n^8} = \frac{1}{3}n^{2-8}(27n^2 + n + 3)$

Step3: Simplify exponent

$\frac{1}{3}n^{-6}(27n^2 + n + 3) = \frac{27n^2 + n + 3}{3n^6} = \frac{9}{n^4} + \frac{1}{3n^5} + \frac{1}{n^6}$

Answer:

1. $7n^5 + 8n^4 + 3n^3$
2. $2n^4 + 8n^2 + \frac{6}{n^4}$
3. $\frac{4}{a^4} + \frac{8}{a^5} + \frac{7}{a^6}$
4. $5x^4 + \frac{8}{x} + \frac{7}{x^2}$
5. $v^2 + 3v + 6$
6. $\frac{9}{n^4} + \frac{1}{3n^5} + \frac{1}{n^6}$