Here's the step-by-step solution for each of the 12 rational expressions. The goal is to
simplify each expression by factoring, canceling common factors, and reducing to lowest terms.
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1) $-\dfrac{36x^3}{42x^2}$
- Factor coefficients: $\dfrac{36}{42} = \dfrac{6 \cdot 6}{6 \cdot 7} = \dfrac{6}{7}$
- Subtract exponents: $x^3 / x^2 = x^{3-2} = x$
- Keep the negative sign.
✔ Answer: $-\dfrac{6x}{7}$
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2) $\dfrac{16r^2}{16r^7}$
- Cancel 16: $\dfrac{r^2}{r^7}$
- Subtract exponents: $r^{2-7} = r^{-5} = \dfrac{1}{r^5}$
✔ Answer: $\dfrac{1}{r^5}$
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3) $\dfrac{16p^2}{28p}$
- Simplify coefficients: $\dfrac{16}{28} = \dfrac{4}{7}$
- Subtract exponents: $p^2 / p = p$
✔ Answer: $\dfrac{4p}{7}$
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4) $\dfrac{32n^2}{24n}$
- Simplify coefficients: $\dfrac{32}{24} = \dfrac{4}{3}$
- Subtract exponents: $n^2 / n = n$
✔ Answer: $\dfrac{4n}{3}$
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5) $\dfrac{70n^2}{28n}$
- Simplify coefficients: $\dfrac{70}{28} = \dfrac{5}{2}$
- Subtract exponents: $n^2 / n = n$
✔ Answer: $\dfrac{5n}{2}$
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6) $\dfrac{15n}{30n^2}$
- Simplify coefficients: $\dfrac{15}{30} = \dfrac{1}{2}$
- Subtract exponents: $n / n^2 = n^{-1} = \dfrac{1}{n}$
✔ Answer: $\dfrac{1}{2n}$
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7) $\dfrac{2r - 4}{r - 2}$
- Factor numerator: $2r - 4 = 2(r - 2)$
- Cancel $(r - 2)$ top and bottom (assuming $r \ne 2$)
✔ Answer: $2$
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8) $\dfrac{45}{10a - 10}$
- Factor denominator: $10a - 10 = 10(a - 1)$
- Simplify: $\dfrac{45}{10(a - 1)} = \dfrac{9}{2(a - 1)}$
✔ Answer: $\dfrac{9}{2(a - 1)}$
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9) $\dfrac{x - 4}{3x^2 - 12x}$
- Factor denominator: $3x^2 - 12x = 3x(x - 4)$
- Cancel $(x - 4)$ top and bottom (assuming $x \ne 4$)
✔ Answer: $\dfrac{1}{3x}$
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10) $\dfrac{15u - 3}{24}$
- Factor numerator: $15u - 3 = 3(5u - 1)$
- Simplify: $\dfrac{3(5u - 1)}{24} = \dfrac{5u - 1}{8}$
✔ Answer: $\dfrac{5u - 1}{8}$
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11) $\dfrac{v - 5}{v^2 - 10v + 25}$
- Factor denominator: $v^2 - 10v + 25 = (v - 5)^2$
- Cancel one $(v - 5)$: $\dfrac{v - 5}{(v - 5)^2} = \dfrac{1}{v - 5}$ (assuming $v \ne 5$)
✔ Answer: $\dfrac{1}{v - 5}$
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12) $\dfrac{x + 6}{x^2 + 5x - 6}$
- Factor denominator: Find two numbers that multiply to -6 and add to 5 → 6 and -1.
So, $x^2 + 5x - 6 = (x + 6)(x - 1)$
- Cancel $(x + 6)$: $\dfrac{x + 6}{(x + 6)(x - 1)} = \dfrac{1}{x - 1}$ (assuming $x \ne -6$)
✔ Answer: $\dfrac{1}{x - 1}$
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##
✔ Final Answers:
1) $-\dfrac{6x}{7}$
2) $\dfrac{1}{r^5}$
3) $\dfrac{4p}{7}$
4) $\dfrac{4n}{3}$
5) $\dfrac{5n}{2}$
6) $\dfrac{1}{2n}$
7) $2$
8) $\dfrac{9}{2(a - 1)}$
9) $\dfrac{1}{3x}$
10) $\dfrac{5u - 1}{8}$
11) $\dfrac{1}{v - 5}$
12) $\dfrac{1}{x - 1}$
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Let me know if you’d like to see any step explained in more detail!
Parent Tip: Review the logic above to help your child master the concept of simplify each expression algebra 1 worksheet.
