Simplifying Expressions Worksheets - Math Monks - Free Printable

Problem: Simplify each expression.

We will solve each expression step by step.

---

#### (1) Simplify:
\[
\frac{1}{x-1} \times \frac{8x-8}{8}
\]

Step 1: Factor the numerator of the second fraction.
\[
8x - 8 = 8(x - 1)
\]
So the expression becomes:
\[
\frac{1}{x-1} \times \frac{8(x-1)}{8}
\]

Step 2: Simplify the constants and cancel out common factors.
\[
\frac{1}{x-1} \times \frac{8(x-1)}{8} = \frac{1}{x-1} \times (x-1) = 1
\]

Final Answer:
\[
\boxed{1}
\]

---

#### (2) Simplify:
\[
\frac{2y-5}{(y+2)(y-3)} \times \frac{y-3}{4y-10}
\]

Step 1: Factor the denominator of the second fraction.
\[
4y - 10 = 2(2y - 5)
\]
So the expression becomes:
\[
\frac{2y-5}{(y+2)(y-3)} \times \frac{y-3}{2(2y-5)}
\]

Step 2: Cancel out common factors.
\[
\frac{2y-5}{(y+2)(y-3)} \times \frac{y-3}{2(2y-5)} = \frac{1}{y+2} \times \frac{1}{2} = \frac{1}{2(y+2)}
\]

Final Answer:
\[
\boxed{\frac{1}{2(y+2)}}
\]

---

#### (3) Simplify:
\[
\frac{3x-9y}{x^2-xy} \div \frac{x^2-9y^2}{x^2-y^2}
\]

Step 1: Rewrite the division as multiplication by the reciprocal.
\[
\frac{3x-9y}{x^2-xy} \div \frac{x^2-9y^2}{x^2-y^2} = \frac{3x-9y}{x^2-xy} \times \frac{x^2-y^2}{x^2-9y^2}
\]

Step 2: Factor each term.
- \(3x - 9y = 3(x - 3y)\)
- \(x^2 - xy = x(x - y)\)
- \(x^2 - 9y^2 = (x - 3y)(x + 3y)\)
- \(x^2 - y^2 = (x - y)(x + y)\)

So the expression becomes:
\[
\frac{3(x-3y)}{x(x-y)} \times \frac{(x-y)(x+y)}{(x-3y)(x+3y)}
\]

Step 3: Cancel out common factors.
\[
\frac{3(x-3y)}{x(x-y)} \times \frac{(x-y)(x+y)}{(x-3y)(x+3y)} = \frac{3}{x} \times \frac{x+y}{x+3y} = \frac{3(x+y)}{x(x+3y)}
\]

Final Answer:
\[
\boxed{\frac{3(x+y)}{x(x+3y)}}
\]

---

#### (4) Simplify:
\[
\frac{r-2}{8r+16} \times \frac{r+2}{r^2-2r}
\]

Step 1: Factor each term.
- \(8r + 16 = 8(r + 2)\)
- \(r^2 - 2r = r(r - 2)\)

So the expression becomes:
\[
\frac{r-2}{8(r+2)} \times \frac{r+2}{r(r-2)}
\]

Step 2: Cancel out common factors.
\[
\frac{r-2}{8(r+2)} \times \frac{r+2}{r(r-2)} = \frac{1}{8} \times \frac{1}{r} = \frac{1}{8r}
\]

Final Answer:
\[
\boxed{\frac{1}{8r}}
\]

---

#### (5) Simplify:
\[
\frac{1}{p-4} \div \frac{3p}{4p-16}
\]

Step 1: Rewrite the division as multiplication by the reciprocal.
\[
\frac{1}{p-4} \div \frac{3p}{4p-16} = \frac{1}{p-4} \times \frac{4p-16}{3p}
\]

Step 2: Factor the numerator of the second fraction.
\[
4p - 16 = 4(p - 4)
\]
So the expression becomes:
\[
\frac{1}{p-4} \times \frac{4(p-4)}{3p}
\]

Step 3: Cancel out common factors.
\[
\frac{1}{p-4} \times \frac{4(p-4)}{3p} = \frac{4}{3p}
\]

Final Answer:
\[
\boxed{\frac{4}{3p}}
\]

---

#### (6) Simplify:
\[
\frac{x^3+2x^2}{y^3-y} \times \frac{y^2-1}{x^2-4}
\]

Step 1: Factor each term.
- \(x^3 + 2x^2 = x^2(x + 2)\)
- \(y^3 - y = y(y^2 - 1) = y(y - 1)(y + 1)\)
- \(y^2 - 1 = (y - 1)(y + 1)\)
- \(x^2 - 4 = (x - 2)(x + 2)\)

So the expression becomes:
\[
\frac{x^2(x+2)}{y(y-1)(y+1)} \times \frac{(y-1)(y+1)}{(x-2)(x+2)}
\]

Step 2: Cancel out common factors.
\[
\frac{x^2(x+2)}{y(y-1)(y+1)} \times \frac{(y-1)(y+1)}{(x-2)(x+2)} = \frac{x^2}{y} \times \frac{1}{x-2} = \frac{x^2}{y(x-2)}
\]

Final Answer:
\[
\boxed{\frac{x^2}{y(x-2)}}
\]

---

#### (7) Simplify:
\[
\frac{5b^2c^2}{10bc} \div \frac{6b^2c}{2c^2}
\]

Step 1: Rewrite the division as multiplication by the reciprocal.
\[
\frac{5b^2c^2}{10bc} \div \frac{6b^2c}{2c^2} = \frac{5b^2c^2}{10bc} \times \frac{2c^2}{6b^2c}
\]

Step 2: Simplify the constants and variables.
\[
\frac{5b^2c^2}{10bc} \times \frac{2c^2}{6b^2c} = \frac{5 \cdot 2 \cdot b^2 \cdot c^2 \cdot c^2}{10 \cdot 6 \cdot b \cdot c \cdot b^2 \cdot c} = \frac{10b^2c^4}{60b^3c^2} = \frac{c^2}{6b}
\]

Final Answer:
\[
\boxed{\frac{c^2}{6b}}
\]

---

#### (8) Simplify:
\[
\frac{2p}{q} \div \frac{4p}{q^2}
\]

Step 1: Rewrite the division as multiplication by the reciprocal.
\[
\frac{2p}{q} \div \frac{4p}{q^2} = \frac{2p}{q} \times \frac{q^2}{4p}
\]

Step 2: Simplify the constants and variables.
\[
\frac{2p}{q} \times \frac{q^2}{4p} = \frac{2p \cdot q^2}{q \cdot 4p} = \frac{2q}{4} = \frac{q}{2}
\]

Final Answer:
\[
\boxed{\frac{q}{2}}
\]

---

Final Answers:

\[
\boxed{1, \frac{1}{2(y+2)}, \frac{3(x+y)}{x(x+3y)}, \frac{1}{8r}, \frac{4}{3p}, \frac{x^2}{y(x-2)}, \frac{c^2}{6b}, \frac{q}{2}}
\]