Simplifying Rational Expressions worksheet for elementary algebra, featuring 18 problems with algebraic fractions to be simplified.

Let’s solve each problem one by one. We’ll simplify rational expressions by factoring numerators and denominators, then canceling common factors.

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1) $\frac{35u}{35u^2}$

Factor numerator and denominator:

Numerator: $35u$
Denominator: $35 \cdot u \cdot u$

Cancel common factor $35u$:

$\frac{35u}{35u^2} = \frac{1}{u}$

✔ Final Answer: $\boxed{\frac{1}{u}}$

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2) $\frac{45v^2}{25x}$

No common variables between numerator and denominator. Simplify numbers:

$\frac{45}{25} = \frac{9}{5}$

So: $\frac{9v^2}{5x}$

✔ Final Answer: $\boxed{\frac{9v^2}{5x}}$

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3) $\frac{x - 8}{x^2 + x - 72}$

Factor denominator: Find two numbers that multiply to -72 and add to 1 → 9 and -8

$x^2 + x - 72 = (x + 9)(x - 8)$

Now expression becomes:

$\frac{x - 8}{(x + 9)(x - 8)}$

Cancel $(x - 8)$:

$\frac{1}{x + 9}$

✔ Final Answer: $\boxed{\frac{1}{x + 9}}$

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4) $\frac{p^2 - 3p - 54}{p - 9}$

Factor numerator: Find two numbers that multiply to -54 and add to -3 → -9 and 6

$p^2 - 3p - 54 = (p - 9)(p + 6)$

Expression becomes:

$\frac{(p - 9)(p + 6)}{p - 9}$

Cancel $(p - 9)$:

$p + 6$

✔ Final Answer: $\boxed{p + 6}$

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5) $\frac{56v - 72}{32v}$

Factor numerator: GCF of 56 and 72 is 8 → $8(7v - 9)$

Denominator: $32v = 8 \cdot 4v$

So: $\frac{8(7v - 9)}{8 \cdot 4v} = \frac{7v - 9}{4v}$

✔ Final Answer: $\boxed{\frac{7v - 9}{4v}}$

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6) $\frac{a + 7}{a^2 + 6a - 7}$

Factor denominator: Two numbers that multiply to -7 and add to 6 → 7 and -1

$a^2 + 6a - 7 = (a + 7)(a - 1)$

Expression becomes:

$\frac{a + 7}{(a + 7)(a - 1)}$

Cancel $(a + 7)$:

$\frac{1}{a - 1}$

✔ Final Answer: $\boxed{\frac{1}{a - 1}}$

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7) $\frac{5v^2 - 5v}{1 - v}$

Factor numerator: $5v(v - 1)$

Note: Denominator is $1 - v = -(v - 1)$

So rewrite:

$\frac{5v(v - 1)}{-(v - 1)} = -5v$

✔ Final Answer: $\boxed{-5v}$

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8) $\frac{2 - x}{x^2 + 4x - 12}$

Factor denominator: Two numbers that multiply to -12 and add to 4 → 6 and -2

$x^2 + 4x - 12 = (x + 6)(x - 2)$

Numerator: $2 - x = -(x - 2)$

So:

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

✔ Final Answer: $\boxed{-\frac{1}{x + 6}}$

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9) $\frac{v^2 + 7v - 30}{9v^2 + 90v}$

Factor numerator: Two numbers that multiply to -30 and add to 7 → 10 and -3

$v^2 + 7v - 30 = (v + 10)(v - 3)$

Factor denominator: $9v(v + 10)$

Expression becomes:

$\frac{(v + 10)(v - 3)}{9v(v + 10)}$

Cancel $(v + 10)$:

$\frac{v - 3}{9v}$

✔ Final Answer: $\boxed{\frac{v - 3}{9v}}$

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10) $\frac{b^2 + b - 30}{3b^2 + 18b}$

Factor numerator: Two numbers that multiply to -30 and add to 1 → 6 and -5

$b^2 + b - 30 = (b + 6)(b - 5)$

Factor denominator: $3b(b + 6)$

Expression becomes:

$\frac{(b + 6)(b - 5)}{3b(b + 6)}$

Cancel $(b + 6)$:

$\frac{b - 5}{3b}$

✔ Final Answer: $\boxed{\frac{b - 5}{3b}}$

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11) $\frac{3x^2 + 5x - 2}{7x^2 + 12x - 4}$

Factor numerator: Use AC method or trial.

Try: $(3x - 1)(x + 2) = 3x^2 + 6x - x - 2 = 3x^2 + 5x - 2$ ✔

Denominator: Try $(7x - 2)(x + 2) = 7x^2 + 14x - 2x - 4 = 7x^2 + 12x - 4$ ✔

So:

$\frac{(3x - 1)(x + 2)}{(7x - 2)(x + 2)}$

Cancel $(x + 2)$:

$\frac{3x - 1}{7x - 2}$

✔ Final Answer: $\boxed{\frac{3x - 1}{7x - 2}}$

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12) $\frac{3b^2 + 26b - 9}{5b^2 + 40b - 45}$

Factor numerator: Try $(3b - 1)(b + 9) = 3b^2 + 27b - b - 9 = 3b^2 + 26b - 9$ ✔

Denominator: Factor out 5 first? Wait, let's factor directly.

Try $(5b - 5)(b + 9)$? No. Let’s use grouping or quadratic formula.

Actually, factor denominator: $5b^2 + 40b - 45 = 5(b^2 + 8b - 9)$

Then factor inside: $b^2 + 8b - 9 = (b + 9)(b - 1)$

So denominator: $5(b + 9)(b - 1)$

Numerator: $(3b - 1)(b + 9)$

Expression:

$\frac{(3b - 1)(b + 9)}{5(b + 9)(b - 1)}$

Cancel $(b + 9)$:

$\frac{3b - 1}{5(b - 1)}$

✔ Final Answer: $\boxed{\frac{3b - 1}{5(b - 1)}}$

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13) $\frac{3a^2 + 21a - 90}{3a^2 + 31a + 10}$

First, factor numerator: Factor out 3? Not necessary yet.

Find factors of $3a^2 + 21a - 90$. First factor out 3:

$3(a^2 + 7a - 30) = 3(a + 10)(a - 3)$

Denominator: $3a^2 + 31a + 10$

Try factoring: Look for two numbers that multiply to 3*10=30 and add to 31 → 30 and 1

Split middle term: $3a^2 + 30a + a + 10 = 3a(a + 10) + 1(a + 10) = (3a + 1)(a + 10)$

So expression:

$\frac{3(a + 10)(a - 3)}{(3a + 1)(a + 10)}$

Cancel $(a + 10)$:

$\frac{3(a - 3)}{3a + 1}$

✔ Final Answer: $\boxed{\frac{3(a - 3)}{3a + 1}}$

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14) $\frac{5m^3 - 57mn + 70n^2}{2m^3 - 16mn - 40n^2}$

Wait — this looks like it might have typos? The terms are mixed degrees: m³, mn, n² — not standard.

But assuming it’s written correctly, perhaps we treat as polynomial in m with coefficients involving n.

Alternatively, maybe it’s meant to be:

Numerator: $5m^2 - 57mn + 70n^2$? But original says m³.

Looking again: Problem 14 is:

$\frac{5m^3 - 57mn + 70n^2}{2m^3 - 16mn - 40n^2}$

This seems odd because you can’t easily factor cubic with mixed terms unless grouped.

Perhaps typo? Maybe it’s supposed to be quadratic in m?

Wait — looking at other problems, likely it’s meant to be:

Actually, checking online or standard worksheets, sometimes these are misprinted.

But let’s assume it’s correct and try to factor.

Alternatively, perhaps group terms? Doesn't seem straightforward.

Wait — another idea: Maybe “mn” is meant to be “m²n”? Or perhaps it’s all in terms of m and n but same degree?

Actually, let me check if numerator and denominator can be factored by treating as quadratics in m, ignoring the cube? That doesn’t make sense.

Wait — perhaps it’s a typo and should be:

Numerator: $5m^2 - 57mn + 70n^2$

Denominator: $2m^2 - 16mn - 40n^2$

That would make more sense for simplifying rational expressions.

Given context of other problems being quadratics, I’ll assume that’s the case.

So revised problem 14:

$\frac{5m^2 - 57mn + 70n^2}{2m^2 - 16mn - 40n^2}$

Factor numerator: Find two numbers that multiply to 5*70=350 and add to -57 → -50 and -7

So: $5m^2 - 50mn - 7mn + 70n^2 = 5m(m - 10n) -7n(m - 10n) = (5m - 7n)(m - 10n)$

Denominator: $2m^2 - 16mn - 40n^2 = 2(m^2 - 8mn - 20n^2)$

Factor inside: $m^2 - 8mn - 20n^2 = (m - 10n)(m + 2n)$

So denominator: $2(m - 10n)(m + 2n)$

Now expression:

$\frac{(5m - 7n)(m - 10n)}{2(m - 10n)(m + 2n)}$

Cancel $(m - 10n)$:

$\frac{5m - 7n}{2(m + 2n)}$

✔ Final Answer: $\boxed{\frac{5m - 7n}{2(m + 2n)}}$

*(Assuming typo corrected from m³ to m²)*

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15) $\frac{v^3 + 11v^2 + 18v}{v^2 + v - 2}$

Factor numerator: Factor out v first: $v(v^2 + 11v + 18)$

Factor quadratic: $v^2 + 11v + 18 = (v + 2)(v + 9)$

So numerator: $v(v + 2)(v + 9)$

Denominator: $v^2 + v - 2 = (v + 2)(v - 1)$

Expression:

$\frac{v(v + 2)(v + 9)}{(v + 2)(v - 1)}$

Cancel $(v + 2)$:

$\frac{v(v + 9)}{v - 1}$

✔ Final Answer: $\boxed{\frac{v(v + 9)}{v - 1}}$

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16) $\frac{2x^3 + 16x^2 + 24x}{x^2 - x - 6}$

Factor numerator: Factor out 2x: $2x(x^2 + 8x + 12)$

Factor quadratic: $x^2 + 8x + 12 = (x + 2)(x + 6)$

So numerator: $2x(x + 2)(x + 6)$

Denominator: $x^2 - x - 6 = (x - 3)(x + 2)$

Expression:

$\frac{2x(x + 2)(x + 6)}{(x - 3)(x + 2)}$

Cancel $(x + 2)$:

$\frac{2x(x + 6)}{x - 3}$

✔ Final Answer: $\boxed{\frac{2x(x + 6)}{x - 3}}$

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17) $\frac{xy + 3x - 2y - 6}{y^2 + y - 6}$

Group numerator: $(xy + 3x) + (-2y - 6) = x(y + 3) -2(y + 3) = (x - 2)(y + 3)$

Denominator: $y^2 + y - 6 = (y + 3)(y - 2)$

Expression:

$\frac{(x - 2)(y + 3)}{(y + 3)(y - 2)}$

Cancel $(y + 3)$:

$\frac{x - 2}{y - 2}$

✔ Final Answer: $\boxed{\frac{x - 2}{y - 2}}$

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18) $\frac{ax - ay + bx - by}{ax - ay - bx + by}$

Group numerator: $(ax - ay) + (bx - by) = a(x - y) + b(x - y) = (a + b)(x - y)$

Denominator: $(ax - ay) - (bx - by) = a(x - y) - b(x - y) = (a - b)(x - y)$

Expression:

$\frac{(a + b)(x - y)}{(a - b)(x - y)}$

Cancel $(x - y)$:

$\frac{a + b}{a - b}$

✔ Final Answer: $\boxed{\frac{a + b}{a - b}}$

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## Final Answers Summary:

1) $\boxed{\frac{1}{u}}$
2) $\boxed{\frac{9v^2}{5x}}$
3) $\boxed{\frac{1}{x + 9}}$
4) $\boxed{p + 6}$
5) $\boxed{\frac{7v - 9}{4v}}$
6) $\boxed{\frac{1}{a - 1}}$
7) $\boxed{-5v}$
8) $\boxed{-\frac{1}{x + 6}}$
9) $\boxed{\frac{v - 3}{9v}}$
10) $\boxed{\frac{b - 5}{3b}}$
11) $\boxed{\frac{3x - 1}{7x - 2}}$
12) $\boxed{\frac{3b - 1}{5(b - 1)}}$
13) $\boxed{\frac{3(a - 3)}{3a + 1}}$
14) $\boxed{\frac{5m - 7n}{2(m + 2n)}}$ *(assuming typo correction)*
15) $\boxed{\frac{v(v + 9)}{v - 1}}$
16) $\boxed{\frac{2x(x + 6)}{x - 3}}$
17) $\boxed{\frac{x - 2}{y - 2}}$
18) $\boxed{\frac{a + b}{a - b}}$