Simplifying Rational Expressions Worksheets - Free Printable

Let's go through each of these rational expressions step by step and simplify them.

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1)

$$
\frac{a^2 + 7a}{a + 3} + \frac{12}{a + 3}
$$

Step 1: Both terms have the same denominator, so combine numerators:
$$
\frac{a^2 + 7a + 12}{a + 3}
$$

Step 2: Factor the numerator:
$$
a^2 + 7a + 12 = (a + 3)(a + 4)
$$

So:
$$
\frac{(a + 3)(a + 4)}{a + 3}
$$

Step 3: Cancel common factor $ a + 3 $ (assuming $ a \ne -3 $):
$$
a + 4
$$

✔ Answer: $ \boxed{a + 4} $

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2)

$$
\frac{13}{4x^2} - \frac{1}{4x^2}
$$

Same denominator, subtract numerators:
$$
\frac{13 - 1}{4x^2} = \frac{12}{4x^2}
$$

Simplify:
$$
\frac{3}{x^2}
$$

✔ Answer: $ \boxed{\frac{3}{x^2}} $

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3)

$$
\frac{6s}{s^2 - 9} + \frac{18}{s^2 - 9}
$$

Same denominator, add numerators:
$$
\frac{6s + 18}{s^2 - 9}
$$

Factor numerator and denominator:

Numerator: $ 6s + 18 = 6(s + 3) $

Denominator: $ s^2 - 9 = (s - 3)(s + 3) $

So:
$$
\frac{6(s + 3)}{(s - 3)(s + 3)}
$$

Cancel $ s + 3 $ (assuming $ s \ne -3 $):
$$
\frac{6}{s - 3}
$$

✔ Answer: $ \boxed{\frac{6}{s - 3}} $

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4)

$$
\frac{3p + 12}{p + 4} + \frac{4 + p}{p + 4}
$$

Same denominator, add numerators:
$$
\frac{3p + 12 + p + 4}{p + 4} = \frac{4p + 16}{p + 4}
$$

Factor numerator:
$$
4p + 16 = 4(p + 4)
$$

So:
$$
\frac{4(p + 4)}{p + 4}
$$

Cancel $ p + 4 $ (assuming $ p \ne -4 $):
$$
4
$$

✔ Answer: $ \boxed{4} $

---

5)

$$
\frac{4(b + 2)}{b + 5} - \frac{(b + 2)}{b + 5}
$$

Same denominator, subtract numerators:
$$
\frac{4(b + 2) - (b + 2)}{b + 5} = \frac{(4 - 1)(b + 2)}{b + 5} = \frac{3(b + 2)}{b + 5}
$$

✔ Answer: $ \boxed{\frac{3(b + 2)}{b + 5}} $

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6)

$$
\frac{17}{5q^3} - \frac{2}{5q^3}
$$

Same denominator, subtract:
$$
\frac{17 - 2}{5q^3} = \frac{15}{5q^3} = \frac{3}{q^3}
$$

✔ Answer: $ \boxed{\frac{3}{q^3}} $

---

7)

$$
\frac{15}{m - 6} + \frac{5(m - 2)}{m - 6}
$$

Same denominator, add numerators:
$$
\frac{15 + 5(m - 2)}{m - 6}
$$

Distribute:
$$
15 + 5m - 10 = 5m + 5
$$

So:
$$
\frac{5m + 5}{m - 6} = \frac{5(m + 1)}{m - 6}
$$

✔ Answer: $ \boxed{\frac{5(m + 1)}{m - 6}} $

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8)

$$
\frac{3n}{n^2} + \frac{4n}{n^2}
$$

Same denominator, add numerators:
$$
\frac{3n + 4n}{n^2} = \frac{7n}{n^2}
$$

Simplify:
$$
\frac{7}{n}
$$

✔ Answer: $ \boxed{\frac{7}{n}} $

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✔ Final Answers:

1) $ a + 4 $
2) $ \frac{3}{x^2} $
3) $ \frac{6}{s - 3} $
4) $ 4 $
5) $ \frac{3(b + 2)}{b + 5} $
6) $ \frac{3}{q^3} $
7) $ \frac{5(m + 1)}{m - 6} $
8) $ \frac{7}{n} $

Let me know if you'd like explanations in a different format or need help with any specific steps!