Simplifying Expressions Worksheets - Math Monks - Free Printable

Let's go through each expression step by step and simplify them. We'll use algebraic techniques such as factoring, canceling common terms, and applying the rules of multiplication and division of rational expressions.

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(1)

$$
\frac{1}{x - 1} \times \frac{8x - 8}{8}
$$

Step 1: Factor numerator of second fraction:
$$
8x - 8 = 8(x - 1)
$$

So,
$$
\frac{1}{x - 1} \times \frac{8(x - 1)}{8}
$$

Step 2: Cancel common terms:
- $8$ in numerator and denominator cancel.
- $(x - 1)$ in numerator and denominator cancel.

Result:
$$
\boxed{1}
$$

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(2)

$$
\frac{2y - 5}{(y + 2)(y - 3)} \times \frac{y - 3}{4y - 10}
$$

Step 1: Factor $4y - 10$:
$$
4y - 10 = 2(2y - 5)
$$

Now rewrite:
$$
\frac{2y - 5}{(y + 2)(y - 3)} \times \frac{y - 3}{2(2y - 5)}
$$

Step 2: Cancel common factors:
- $2y - 5$ cancels.
- $y - 3$ cancels.

Left with:
$$
\frac{1}{(y + 2)} \times \frac{1}{2} = \frac{1}{2(y + 2)}
$$

Answer:
$$
\boxed{\frac{1}{2(y + 2)}}
$$

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(3)

$$
\frac{3x - 9y}{x^2 - xy} \div \frac{x^2 - 9y^2}{x^2 - y^2}
$$

Step 1: Rewrite division as multiplication by reciprocal:
$$
\frac{3x - 9y}{x^2 - xy} \times \frac{x^2 - y^2}{x^2 - 9y^2}
$$

Step 2: Factor all expressions:

- $3x - 9y = 3(x - 3y)$
- $x^2 - xy = x(x - y)$
- $x^2 - y^2 = (x - y)(x + y)$
- $x^2 - 9y^2 = (x - 3y)(x + 3y)$

Now substitute:
$$
\frac{3(x - 3y)}{x(x - y)} \times \frac{(x - y)(x + y)}{(x - 3y)(x + 3y)}
$$

Step 3: Cancel common factors:
- $x - 3y$: cancels
- $x - y$: cancels

Left with:
$$
\frac{3}{x} \times \frac{x + y}{x + 3y} = \frac{3(x + y)}{x(x + 3y)}
$$

Answer:
$$
\boxed{\frac{3(x + y)}{x(x + 3y)}}
$$

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(4)

$$
\frac{r - 2}{8r + 16} \times \frac{r + 2}{r^2 - 2r}
$$

Step 1: Factor denominators and numerators:

- $8r + 16 = 8(r + 2)$
- $r^2 - 2r = r(r - 2)$

Now:
$$
\frac{r - 2}{8(r + 2)} \times \frac{r + 2}{r(r - 2)}
$$

Step 2: Cancel common terms:
- $r - 2$: cancels
- $r + 2$: cancels

Left with:
$$
\frac{1}{8} \times \frac{1}{r} = \frac{1}{8r}
$$

Answer:
$$
\boxed{\frac{1}{8r}}
$$

---

(5)

$$
\frac{1}{p - 4} \div \frac{3p}{4p - 16}
$$

Step 1: Rewrite as multiplication:
$$
\frac{1}{p - 4} \times \frac{4p - 16}{3p}
$$

Step 2: Factor $4p - 16 = 4(p - 4)$

So:
$$
\frac{1}{p - 4} \times \frac{4(p - 4)}{3p}
$$

Step 3: Cancel $p - 4$:
$$
\frac{1}{1} \times \frac{4}{3p} = \frac{4}{3p}
$$

Answer:
$$
\boxed{\frac{4}{3p}}
$$

---

(6)

$$
\frac{x^3 + 2x^2}{y^3 - y} \times \frac{y^2 - 1}{x^2 - 4}
$$

Step 1: Factor all expressions:

- $x^3 + 2x^2 = x^2(x + 2)$
- $y^3 - y = y(y^2 - 1) = y(y - 1)(y + 1)$
- $y^2 - 1 = (y - 1)(y + 1)$
- $x^2 - 4 = (x - 2)(x + 2)$

Now:
$$
\frac{x^2(x + 2)}{y(y - 1)(y + 1)} \times \frac{(y - 1)(y + 1)}{(x - 2)(x + 2)}
$$

Step 2: Cancel common factors:
- $x + 2$: cancels
- $y - 1$: cancels
- $y + 1$: cancels

Left with:
$$
\frac{x^2}{y} \times \frac{1}{x - 2} = \frac{x^2}{y(x - 2)}
$$

Answer:
$$
\boxed{\frac{x^2}{y(x - 2)}}
$$

---

(7)

$$
\frac{5b^2c^2}{10bc} \div \frac{6b^2c}{2c^2}
$$

Step 1: Rewrite as multiplication:
$$
\frac{5b^2c^2}{10bc} \times \frac{2c^2}{6b^2c}
$$

Step 2: Simplify each fraction first:

- $\frac{5b^2c^2}{10bc} = \frac{1}{2} \cdot b^{2-1} c^{2-1} = \frac{1}{2}bc$
- $\frac{2c^2}{6b^2c} = \frac{1}{3} \cdot \frac{c^{2-1}}{b^2} = \frac{c}{3b^2}$

Now multiply:
$$
\left(\frac{1}{2}bc\right) \times \left(\frac{c}{3b^2}\right) = \frac{1}{2} \cdot \frac{1}{3} \cdot b^{1-2} \cdot c^{1+1} = \frac{1}{6} \cdot b^{-1} \cdot c^2 = \frac{c^2}{6b}
$$

Alternatively, do it directly without simplifying first:

$$
\frac{5b^2c^2}{10bc} \times \frac{2c^2}{6b^2c} = \frac{5 \cdot 2 \cdot b^2c^2 \cdot c^2}{10 \cdot 6 \cdot bc \cdot b^2c}
= \frac{10 b^2 c^4}{60 b^3 c^2}
= \frac{1}{6} \cdot b^{-1} \cdot c^{2} = \frac{c^2}{6b}
$$

Answer:
$$
\boxed{\frac{c^2}{6b}}
$$

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(8)

$$
\frac{2p}{q} \div \frac{4p}{q^2}
$$

Step 1: Rewrite as multiplication:
$$
\frac{2p}{q} \times \frac{q^2}{4p}
$$

Step 2: Cancel common terms:
- $2p$ and $4p$: $2p / 4p = 1/2$
- $q$ and $q^2$: $q^2 / q = q$

So:
$$
\frac{1}{2} \cdot q = \frac{q}{2}
$$

Answer:
$$
\boxed{\frac{q}{2}}
$$

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✔ Final Answers:

1. $ \boxed{1} $
2. $ \boxed{\frac{1}{2(y + 2)}} $
3. $ \boxed{\frac{3(x + y)}{x(x + 3y)}} $
4. $ \boxed{\frac{1}{8r}} $
5. $ \boxed{\frac{4}{3p}} $
6. $ \boxed{\frac{x^2}{y(x - 2)}} $
7. $ \boxed{\frac{c^2}{6b}} $
8. $ \boxed{\frac{q}{2}} $

Let me know if you'd like these explained further!