11) $$ \frac{3 a b \cdot 2 a^{4}}{4 a^{2} b^{3}} $$ $$ \begin{array}{ll} ext { 13) } \frac{2 y x^{4} \cdot 2 x}{3 x^{3} y^{2}} & ext { 12) } \frac{4 x y^{3}}{x y^{2} \cdot 2 x^{3} y^{3}} \ ext { 15) } \frac{2 x^{4} y^{3}}{\left(x^{-2} y^{2} ight)^{4}} & ext { 14) } \frac{3 x^{2}}{4 y x^{3} \cdot 2 y x^{3} \cdot y x^{4}} \ ext { 17) }\left(\frac{x^{-4} y^{2}}{x^{0}} ight)^{3} & ext { 16) } \frac{2 n^{-3}}{\left(2 m^{4} ight)^{-2}}\end{array} $$

Question

11) $$ \frac{3 a b \cdot 2 a^{4}}{4 a^{2} b^{3}} $$ $$ \begin{array}{ll}	ext { 13) } \frac{2 y x^{4} \cdot 2 x}{3 x^{3} y^{2}} & 	ext { 12) } \frac{4 x y^{3}}{x y^{2} \cdot 2 x^{3} y^{3}} \ 	ext { 15) } \frac{2 x^{4} y^{3}}{\left(x^{-2} y^{2}ight)^{4}} & 	ext { 14) } \frac{3 x^{2}}{4 y x^{3} \cdot 2 y x^{3} \cdot y x^{4}} \ 	ext { 17) }\left(\frac{x^{-4} y^{2}}{x^{0}}ight)^{3} & 	ext { 16) } \frac{2 n^{-3}}{\left(2 m^{4}ight)^{-2}}\end{array} $$

UpStudy AI · Accepted Answer

Sure! Let's simplify each of the given algebraic expressions step by step.

---

### **Problem 11**
$$
\frac{3ab \cdot 2a^{4}}{4a^{2}b^{3}}
$$

**Solution:**
1. **Multiply the numerators:**
   $$
   3ab \cdot 2a^{4} = 6a^{1+4}b = 6a^{5}b
   $$
   
2. **Write the denominator:**
   $$
   4a^{2}b^{3}
   $$
   
3. **Combine the fraction:**
   $$
   \frac{6a^{5}b}{4a^{2}b^{3}} = \frac{6}{4} \cdot \frac{a^{5}}{a^{2}} \cdot \frac{b}{b^{3}}
   $$
   
4. **Simplify the coefficients and exponents:**
   $$
   \frac{6}{4} = \frac{3}{2}, \quad a^{5-2} = a^{3}, \quad b^{1-3} = b^{-2} = \frac{1}{b^{2}}
   $$
   
5. **Final simplified form:**
   $$
   \frac{3a^{3}}{2b^{2}}
   $$

---

### **Problem 12**
$$
\frac{4xy^{3}}{xy^{2} \cdot 2x^{3}y^{3}}
$$

**Solution:**
1. **Multiply the denominators:**
   $$
   xy^{2} \cdot 2x^{3}y^{3} = 2x^{1+3}y^{2+3} = 2x^{4}y^{5}
   $$
   
2. **Write the fraction:**
   $$
   \frac{4xy^{3}}{2x^{4}y^{5}}
   $$
   
3. **Simplify the coefficients and exponents:**
   $$
   \frac{4}{2} = 2, \quad x^{1-4} = x^{-3} = \frac{1}{x^{3}}, \quad y^{3-5} = y^{-2} = \frac{1}{y^{2}}
   $$
   
4. **Final simplified form:**
   $$
   \frac{2}{x^{3}y^{2}} \quad \text{or} \quad 2x^{-3}y^{-2}
   $$

---

### **Problem 13**
$$
\frac{2yx^{4} \cdot 2x}{3x^{3}y^{2}}
$$

**Solution:**
1. **Multiply the numerators:**
   $$
   2yx^{4} \cdot 2x = 4yx^{4+1} = 4yx^{5}
   $$
   
2. **Write the denominator:**
   $$
   3x^{3}y^{2}
   $$
   
3. **Combine the fraction:**
   $$
   \frac{4yx^{5}}{3x^{3}y^{2}} = \frac{4}{3} \cdot \frac{y}{y^{2}} \cdot \frac{x^{5}}{x^{3}}
   $$
   
4. **Simplify the coefficients and exponents:**
   $$
   \frac{4}{3}, \quad y^{1-2} = y^{-1} = \frac{1}{y}, \quad x^{5-3} = x^{2}
   $$
   
5. **Final simplified form:**
   $$
   \frac{4x^{2}}{3y} \quad \text{or} \quad \frac{4}{3}x^{2}y^{-1}
   $$

---

### **Problem 14**
$$
\frac{3x^{2}}{4yx^{3} \cdot 2yx^{3} \cdot yx^{4}}
$$

**Solution:**
1. **Multiply the denominators:**
   $$
   4yx^{3} \cdot 2yx^{3} \cdot yx^{4} = (4 \cdot 2 \cdot 1) \cdot y^{1+1+1} \cdot x^{3+3+4} = 8y^{3}x^{10}
   $$
   
2. **Write the fraction:**
   $$
   \frac{3x^{2}}{8y^{3}x^{10}}
   $$
   
3. **Simplify the coefficients and exponents:**
   $$
   \frac{3}{8}, \quad x^{2-10} = x^{-8} = \frac{1}{x^{8}}, \quad y^{-3} = \frac{1}{y^{3}}
   $$
   
4. **Final simplified form:**
   $$
   \frac{3}{8x^{8}y^{3}} \quad \text{or} \quad 3 \cdot 8^{-1}x^{-8}y^{-3}
   $$

---

### **Problem 15**
$$
\frac{2x^{4}y^{3}}{(x^{-2}y^{2})^{4}}
$$

**Solution:**
1. **Simplify the denominator using exponents:**
   $$
   (x^{-2}y^{2})^{4} = x^{-2 \cdot 4}y^{2 \cdot 4} = x^{-8}y^{8}
   $$
   
2. **Write the fraction:**
   $$
   \frac{2x^{4}y^{3}}{x^{-8}y^{8}} = 2x^{4 - (-8)}y^{3 - 8} = 2x^{12}y^{-5}
   $$
   
3. **Express with positive exponents:**
   $$
   2x^{12} \cdot \frac{1}{y^{5}} = \frac{2x^{12}}{y^{5}}
   $$
   
4. **Final simplified form:**
   $$
   \frac{2x^{12}}{y^{5}}
   $$

---

### **Problem 16**
$$
\frac{2n^{-3}}{(2m^{4})^{-2}}
$$

**Solution:**
1. **Simplify the denominator:**
   $$
   (2m^{4})^{-2} = 2^{-2}m^{4 \cdot (-2)} = \frac{1}{4}m^{-8}
   $$
   
2. **Rewrite the fraction:**
   $$
   \frac{2n^{-3}}{\frac{1}{4}m^{-8}} = 2n^{-3} \cdot \frac{4}{1}m^{8} = 8m^{8}n^{-3}
   $$
   
3. **Express with positive exponents:**
   $$
   8m^{8} \cdot \frac{1}{n^{3}} = \frac{8m^{8}}{n^{3}}
   $$
   
4. **Final simplified form:**
   $$
   \frac{8m^{8}}{n^{3}}
   $$

---

### **Problem 17**
$$
\left(\frac{x^{-4}y^{2}}{x^{0}}\right)^{3}
$$

**Solution:**
1. **Simplify inside the parentheses:**
   $$
   x^{0} = 1 \quad \Rightarrow \quad \frac{x^{-4}y^{2}}{1} = x^{-4}y^{2}
   $$
   
2. **Raise to the power of 3:**
   $$
   (x^{-4}y^{2})^{3} = x^{-4 \cdot 3}y^{2 \cdot 3} = x^{-12}y^{6}
   $$
   
3. **Express with positive exponents:**
   $$
   \frac{y^{6}}{x^{12}}
   $$
   
4. **Final simplified form:**
   $$
   \frac{y^{6}}{x^{12}}
   $$

---

### **Summary of Simplified Expressions:**

1. **Problem 11:** $$\displaystyle \frac{3a^{3}}{2b^{2}}$$
2. **Problem 12:** $$\displaystyle \frac{2}{x^{3}y^{2}}$$
3. **Problem 13:** $$\displaystyle \frac{4x^{2}}{3y}$$
4. **Problem 14:** $$\displaystyle \frac{3}{8x^{8}y^{3}}$$
5. **Problem 15:** $$\displaystyle \frac{2x^{12}}{y^{5}}$$
6. **Problem 16:** $$\displaystyle \frac{8m^{8}}{n^{3}}$$
7. **Problem 17:** $$\displaystyle \frac{y^{6}}{x^{12}}$$

Feel free to ask if you have any further questions or need additional explanations!
Here are the simplified forms of the given algebraic expressions: 1. **Problem 11:** $$\frac{3a^{3}}{2b^{2}}$$ 2. **Problem 12:** $$\frac{2}{x^{3}y^{2}}$$ 3. **Problem 13:** $$\frac{4x^{2}}{3y}$$ 4. **Problem 14:** $$\frac{3}{8x^{8}y^{3}}$$ 5. **Problem 15:** $$\frac{2x^{12}}{y^{5}}$$ 6. **Problem 16:** $$\frac{8m^{8}}{n^{3}}$$ 7. **Problem 17:** $$\frac{y^{6}}{x^{12}}$$

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