\begin{tabular}{l} A. Aply the laws of exponents to simplify the given exponential expressions. Show your comp \ Highlight the final answer. $$ \left(4 x y^{2} ight) \cdot\left(7 x^{3} y^{4} ight) $$ \ b. $$ \left(-3 a^{4} b^{5} ight)^{4} $$ \ c. $$ \left(-\frac{1}{2} ight)^{2} \cdot\left(-\frac{1}{2} ight)^{4} $$ \ d. $$ \frac{48 m^{12} n^{9}}{4 m^{9} n} $$ \ \hline\end{tabular}

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\begin{tabular}{l} A. Aply the laws of exponents to simplify the given exponential expressions. Show your comp \ Highlight the final answer. $$ \left(4 x y^{2}ight) \cdot\left(7 x^{3} y^{4}ight) $$ \ b. $$ \left(-3 a^{4} b^{5}ight)^{4} $$ \ c. $$ \left(-\frac{1}{2}ight)^{2} \cdot\left(-\frac{1}{2}ight)^{4} $$ \ d. $$ \frac{48 m^{12} n^{9}}{4 m^{9} n} $$ \ \hline\end{tabular}

UpStudy AI · Accepted Answer

**A. Apply the laws of exponents to simplify the given exponential expressions. Show your computations. Highlight the final answer.**

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### a. $$ \left(4 x y^{2}\right) \cdot\left(7 x^{3} y^{4}\right) $$

**Solution:**

1. **Multiply the coefficients:**
   $$
   4 \times 7 = 28
   $$
   
2. **Apply the law of exponents for the same base:**
   $$
   x^{1} \times x^{3} = x^{1+3} = x^{4}
   $$
   $$
   y^{2} \times y^{4} = y^{2+4} = y^{6}
   $$

3. **Combine the results:**
   $$
   28 \times x^{4} \times y^{6} = 28x^{4}y^{6}
   $$

**Final Answer:**  
$$
\boxed{28x^{4}y^{6}}
$$

---

### b. $$ \left(-3 a^{4} b^{5}\right)^{4} $$

**Solution:**

1. **Apply the power of a product rule:**
   $$
   \left(-3 a^{4} b^{5}\right)^{4} = (-3)^{4} \times (a^{4})^{4} \times (b^{5})^{4}
   $$
   
2. **Calculate each part separately:**
   $$
   (-3)^{4} = 81
   $$
   $$
   (a^{4})^{4} = a^{16}
   $$
   $$
   (b^{5})^{4} = b^{20}
   $$
   
3. **Combine the results:**
   $$
   81 \times a^{16} \times b^{20} = 81a^{16}b^{20}
   $$

**Final Answer:**  
$$
\boxed{81a^{16}b^{20}}
$$

---

### c. $$ \left(-\frac{1}{2}\right)^{2} \cdot\left(-\frac{1}{2}\right)^{4} $$

**Solution:**

1. **Apply the law of exponents for the same base:**
   $$
   \left(-\frac{1}{2}\right)^{2} \times \left(-\frac{1}{2}\right)^{4} = \left(-\frac{1}{2}\right)^{2+4} = \left(-\frac{1}{2}\right)^{6}
   $$
   
2. **Simplify the expression:**
   $$
   \left(-\frac{1}{2}\right)^{6} = \left(\frac{1}{2}\right)^{6} = \frac{1}{64}
   $$
   (Note: Even exponent makes the negative sign positive)

**Final Answer:**  
$$
\boxed{\frac{1}{64}}
$$

---

### d. $$ \frac{48 m^{12} n^{9}}{4 m^{9} n} $$

**Solution:**

1. **Simplify the coefficients:**
   $$
   \frac{48}{4} = 12
   $$
   
2. **Apply the law of exponents for division:**
   $$
   \frac{m^{12}}{m^{9}} = m^{12-9} = m^{3}
   $$
   $$
   \frac{n^{9}}{n^{1}} = n^{9-1} = n^{8}
   $$
   
3. **Combine the results:**
   $$
   12 \times m^{3} \times n^{8} = 12m^{3}n^{8}
   $$

**Final Answer:**  
$$
\boxed{12m^{3}n^{8}}
$$

---
**Simplified Answers:** a. $$ 28x^{4}y^{6} $$ b. $$ 81a^{16}b^{20} $$ c. $$ \frac{1}{64} $$ d. $$ 12m^{3}n^{8} $$

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