Question

kuta software - infinite algebra 1
simplifying rational expressions
simplify each expression.

1. $-\frac{36x^3}{42x^2}$
2. $\frac{16r^2}{16r^3}$
3. $\frac{16p^2}{28p}$
4. $\frac{32n^2}{24n}$
5. $-\frac{70n^2}{28n}$
6. $\frac{15n}{30n^3}$
7. $\frac{2r - 4}{r - 2}$
8. $\frac{45}{10a - 10}$
9. $\frac{x - 4}{3x^2 - 12x}$
10. $\frac{15a - 3}{24}$
11. $\frac{v - 5}{v^2 - 10v + 25}$
12. $\frac{x + 6}{x^2 + 5x - 6}$

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Explanation:

Step1: Simplify coefficients and variables

$-\frac{36x^3}{42x^2} = -\frac{36}{42} \cdot x^{3-2} = -\frac{6x}{7}$

Step2: Simplify coefficients and variables

$\frac{16r^2}{16r^3} = \frac{16}{16} \cdot r^{2-3} = \frac{1}{r}$

Step3: Simplify coefficients and variables

$\frac{16p^2}{28p} = \frac{16}{28} \cdot p^{2-1} = \frac{4p}{7}$

Step4: Simplify coefficients and variables

$\frac{32n^2}{24n} = \frac{32}{24} \cdot n^{2-1} = \frac{4n}{3}$

Step5: Simplify coefficients and variables

$-\frac{70n^2}{28n} = -\frac{70}{28} \cdot n^{2-1} = -\frac{5n}{2}$

Step6: Simplify coefficients and variables

$\frac{15n}{30n^3} = \frac{15}{30} \cdot n^{1-3} = \frac{1}{2n^2}$

Step7: Factor numerator, cancel terms

$\frac{2r-4}{r-2} = \frac{2(r-2)}{r-2} = 2$

Step8: Factor denominator, simplify fraction

$\frac{45}{10a-10} = \frac{45}{10(a-1)} = \frac{9}{2(a-1)}$

Step9: Factor denominator, cancel terms

$\frac{x-4}{3x^2-12x} = \frac{x-4}{3x(x-4)} = \frac{1}{3x}$

Step10: Factor numerator, simplify fraction

$\frac{15a-3}{24} = \frac{3(5a-1)}{24} = \frac{5a-1}{8}$

Step11: Factor denominator, cancel terms

$\frac{v-5}{v^2-10v+25} = \frac{v-5}{(v-5)^2} = \frac{1}{v-5}$

Step12: Factor denominator, cancel terms

$\frac{x+6}{x^2+5x-6} = \frac{x+6}{(x+6)(x-1)} = \frac{1}{x-1}$

Answer:

1. $-\frac{6x}{7}$
2. $\frac{1}{r}$
3. $\frac{4p}{7}$
4. $\frac{4n}{3}$
5. $-\frac{5n}{2}$
6. $\frac{1}{2n^2}$
7. $2$
8. $\frac{9}{2(a-1)}$
9. $\frac{1}{3x}$
10. $\frac{5a-1}{8}$
11. $\frac{1}{v-5}$
12. $\frac{1}{x-1}$