Math worksheet for simplifying algebraic expressions, featuring eight problems with fractions and variables.

Let's solve each expression step by step and simplify them.

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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 the expression becomes:
$$
\frac{1}{x - 1} \times \frac{8(x - 1)}{8}
$$

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

$$
= 1
$$

✔ Answer: $1$

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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 out.
- $y - 3$ cancels out.

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

✔ Answer: $\frac{1}{2(y + 2)}$

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

Step 1: Convert division to multiplication:
$$
\frac{3x - 9y}{x^2 - xy} \times \frac{x^2 - y^2}{x^2 - 9y^2}
$$

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 terms:
- $(x - 3y)$ cancels
- $(x - y)$ cancels

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

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

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

Step 1: Factor denominators:

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

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

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

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

✔ Answer: $\frac{1}{8r}$

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

Step 1: Convert to 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: $\frac{4}{3p}$

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

Step 1: Factor each expression:

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

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

Step 2: Cancel common terms:
- $(y - 1)$: cancels
- $(y + 1)$: cancels
- $(x + 2)$: cancels

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

✔ Answer: $\frac{x^2}{y(x - 2)}$

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

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

Step 2: Simplify each fraction first:

- $\frac{5b^2c^2}{10bc} = \frac{1}{2} \cdot \frac{b^2c^2}{bc} = \frac{1}{2} \cdot bc = \frac{bc}{2}$
- But let’s keep it symbolic for cancellation.

Better to write both together:

$$
\frac{5b^2c^2}{10bc} \times \frac{2c^2}{6b^2c} = \frac{5 \cdot b^2 \cdot c^2 \cdot 2 \cdot c^2}{10 \cdot b \cdot c \cdot 6 \cdot b^2 \cdot c}
$$

Simplify numerator and denominator:

Numerator: $5 \cdot 2 \cdot b^2 \cdot c^2 \cdot c^2 = 10b^2c^4$

Denominator: $10 \cdot 6 \cdot b \cdot b^2 \cdot c \cdot c = 60b^3c^2$

So:
$$
\frac{10b^2c^4}{60b^3c^2} = \frac{1}{6} \cdot \frac{c^{4-2}}{b^{3-2}} = \frac{1}{6} \cdot \frac{c^2}{b} = \frac{c^2}{6b}
$$

✔ Answer: $\frac{c^2}{6b}$

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(8) $\frac{2p}{q} \div \frac{4p}{q^2}$

Step 1: Convert to multiplication:
$$
\frac{2p}{q} \times \frac{q^2}{4p}
$$

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

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

✔ Answer: $\frac{q}{2}$

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✔ Final Answers:

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

Let me know if you'd like these written neatly on a printable sheet!