Question: Simplify each expression.
1) $\frac{-36x^3}{42x^2}$
2) $\frac{16r^2}{16r^3}$
3) $\frac{16p^2}{28p}$
4) $\frac{32n^2}{24n}$
5) $\frac{-70n^2}{28n}$
6) $\frac{15n}{30n^3}$
7) $\frac{2r - 4}{r - 2}$
8) $\frac{45}{10a - 10}$
9) $\frac{x - 4}{3x^2 - 12x}$
10) $\frac{15a - 3}{24}$
11) $\frac{v - 5}{v^2 - 10v + 25}$
12) $\frac{x + 6}{x^2 + 5x - 6}$
1. **Problem:** Simplify $\frac{-36x^3}{42x^2}$.
**Step 1:** Write the expression:
$$\frac{-36x^3}{42x^2}$$
**Step 2:** Factor numerator and denominator:
$$\frac{-36 \cdot x^3}{42 \cdot x^2}$$
**Step 3:** Simplify coefficients by dividing numerator and denominator by their greatest common divisor (GCD) 6:
$$\frac{\cancel{6} \cdot -6 \cdot x^3}{\cancel{6} \cdot 7 \cdot x^2} = \frac{-6x^3}{7x^2}$$
**Step 4:** Simplify powers of $x$ by subtracting exponents:
$$\frac{-6 \cdot \cancel{x^2} \cdot x}{7 \cdot \cancel{x^2}} = \frac{-6x}{7}$$
**Answer:** $\boxed{\frac{-6x}{7}}$
2. **Problem:** Simplify $\frac{16r^2}{16r^3}$.
**Step 1:** Write the expression:
$$\frac{16r^2}{16r^3}$$
**Step 2:** Cancel common factor 16:
$$\frac{\cancel{16}r^2}{\cancel{16}r^3} = \frac{r^2}{r^3}$$
**Step 3:** Simplify powers of $r$ by subtracting exponents:
$$\frac{\cancel{r^2}}{r^{3-2} \cdot \cancel{r^2}} = \frac{1}{r}$$
**Answer:** $\boxed{\frac{1}{r}}$
3. **Problem:** Simplify $\frac{16p^2}{28p}$.
**Step 1:** Write the expression:
$$\frac{16p^2}{28p}$$
**Step 2:** Simplify coefficients by dividing numerator and denominator by GCD 4:
$$\frac{\cancel{4} \cdot 4 p^2}{\cancel{4} \cdot 7 p} = \frac{4p^2}{7p}$$
**Step 3:** Simplify powers of $p$ by subtracting exponents:
$$\frac{4 \cdot \cancel{p} \cdot p}{7 \cdot \cancel{p}} = \frac{4p}{7}$$
**Answer:** $\boxed{\frac{4p}{7}}$
4. **Problem:** Simplify $\frac{32n^2}{24n}$.
**Step 1:** Write the expression:
$$\frac{32n^2}{24n}$$
**Step 2:** Simplify coefficients by dividing numerator and denominator by GCD 8:
$$\frac{\cancel{8} \cdot 4 n^2}{\cancel{8} \cdot 3 n} = \frac{4n^2}{3n}$$
**Step 3:** Simplify powers of $n$ by subtracting exponents:
$$\frac{4 \cdot \cancel{n} \cdot n}{3 \cdot \cancel{n}} = \frac{4n}{3}$$
**Answer:** $\boxed{\frac{4n}{3}}$
5. **Problem:** Simplify $\frac{-70n^2}{28n}$.
**Step 1:** Write the expression:
$$\frac{-70n^2}{28n}$$
**Step 2:** Simplify coefficients by dividing numerator and denominator by GCD 14:
$$\frac{\cancel{14} \cdot -5 n^2}{\cancel{14} \cdot 2 n} = \frac{-5n^2}{2n}$$
**Step 3:** Simplify powers of $n$ by subtracting exponents:
$$\frac{-5 \cdot \cancel{n} \cdot n}{2 \cdot \cancel{n}} = \frac{-5n}{2}$$
**Answer:** $\boxed{\frac{-5n}{2}}$
6. **Problem:** Simplify $\frac{15n}{30n^3}$.
**Step 1:** Write the expression:
$$\frac{15n}{30n^3}$$
**Step 2:** Simplify coefficients by dividing numerator and denominator by GCD 15:
$$\frac{\cancel{15} n}{\cancel{15} 2 n^3} = \frac{n}{2 n^3}$$
**Step 3:** Simplify powers of $n$ by subtracting exponents:
$$\frac{\cancel{n}}{2 n^{3-1} \cdot \cancel{n}} = \frac{1}{2 n^2}$$
**Answer:** $\boxed{\frac{1}{2 n^2}}$
7. **Problem:** Simplify $\frac{2r - 4}{r - 2}$.
**Step 1:** Factor numerator:
$$2r - 4 = 2(r - 2)$$
**Step 2:** Write expression:
$$\frac{2(r - 2)}{r - 2}$$
**Step 3:** Cancel common factor $r - 2$:
$$\frac{2 \cancel{(r - 2)}}{\cancel{r - 2}} = 2$$
**Answer:** $\boxed{2}$
8. **Problem:** Simplify $\frac{45}{10a - 10}$.
**Step 1:** Factor denominator:
$$10a - 10 = 10(a - 1)$$
**Step 2:** Write expression:
$$\frac{45}{10(a - 1)}$$
**Step 3:** Simplify coefficients by dividing numerator and denominator by 5:
$$\frac{\cancel{5} \cdot 9}{\cancel{5} \cdot 2 (a - 1)} = \frac{9}{2(a - 1)}$$
**Answer:** $\boxed{\frac{9}{2(a - 1)}}$
9. **Problem:** Simplify $\frac{x - 4}{3x^2 - 12x}$.
**Step 1:** Factor denominator:
$$3x^2 - 12x = 3x(x - 4)$$
**Step 2:** Write expression:
$$\frac{x - 4}{3x(x - 4)}$$
**Step 3:** Cancel common factor $x - 4$:
$$\frac{\cancel{x - 4}}{3x \cdot \cancel{x - 4}} = \frac{1}{3x}$$
**Answer:** $\boxed{\frac{1}{3x}}$
10. **Problem:** Simplify $\frac{15a - 3}{24}$.
**Step 1:** Factor numerator:
$$15a - 3 = 3(5a - 1)$$
**Step 2:** Write expression:
$$\frac{3(5a - 1)}{24}$$
**Step 3:** Simplify coefficients by dividing numerator and denominator by 3:
$$\frac{\cancel{3}(5a - 1)}{8 \cdot \cancel{3}} = \frac{5a - 1}{8}$$
**Answer:** $\boxed{\frac{5a - 1}{8}}$
11. **Problem:** Simplify $\frac{v - 5}{v^2 - 10v + 25}$.
**Step 1:** Factor denominator:
$$v^2 - 10v + 25 = (v - 5)^2$$
**Step 2:** Write expression:
$$\frac{v - 5}{(v - 5)^2}$$
**Step 3:** Simplify by canceling one $v - 5$ factor:
$$\frac{\cancel{v - 5}}{(v - 5) \cdot \cancel{v - 5}} = \frac{1}{v - 5}$$
**Answer:** $\boxed{\frac{1}{v - 5}}$
12. **Problem:** Simplify $\frac{x + 6}{x^2 + 5x - 6}$.
**Step 1:** Factor denominator:
$$x^2 + 5x - 6 = (x + 6)(x - 1)$$
**Step 2:** Write expression:
$$\frac{x + 6}{(x + 6)(x - 1)}$$
**Step 3:** Cancel common factor $x + 6$:
$$\frac{\cancel{x + 6}}{\cancel{x + 6} (x - 1)} = \frac{1}{x - 1}$$
**Answer:** $\boxed{\frac{1}{x - 1}}$