$$ \begin{array}{ll} ext { (b) }\left(-2 x^{3} y^{2} ight)^{3} & ext { (c) }\left(3^{3} ight)^{2} \cdot 3^{2} \ ext { (e) }\left(\frac{-12 x^{3}}{15 y^{-2}} ight)^{2} & ext { (f) }\left(\frac{10 y^{4}}{4 y^{6}} ight)^{-2}\end{array} $$

Question

$$ \begin{array}{ll}	ext { (b) }\left(-2 x^{3} y^{2}ight)^{3} & 	ext { (c) }\left(3^{3}ight)^{2} \cdot 3^{2} \ 	ext { (e) }\left(\frac{-12 x^{3}}{15 y^{-2}}ight)^{2} & 	ext { (f) }\left(\frac{10 y^{4}}{4 y^{6}}ight)^{-2}\end{array} $$

UpStudy AI · Accepted Answer

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

---

### **(b) $$\left(-2 x^{3} y^{2}\right)^{3}$$**

**Step-by-Step Simplification:**

1. **Apply the exponent to each factor inside the parentheses:**
   $$
   \left(-2\right)^3 \cdot \left(x^{3}\right)^3 \cdot \left(y^{2}\right)^3
   $$

2. **Calculate each part:**
   - $$\left(-2\right)^3 = -8$$
   - $$\left(x^{3}\right)^3 = x^{9}$$ (since $$3 \times 3 = 9$$)
   - $$\left(y^{2}\right)^3 = y^{6}$$ (since $$2 \times 3 = 6$$)

3. **Combine the results:**
   $$
   -8 x^{9} y^{6}
   $$

**Final Answer:**
$$
-8 x^{9} y^{6}
$$

---

### **(c) $$\left(3^{3}\right)^{2} \cdot 3^{2}$$**

**Step-by-Step Simplification:**

1. **Simplify $$\left(3^{3}\right)^{2}$$:**
   $$
   3^{3 \times 2} = 3^{6}
   $$

2. **Multiply by $$3^{2}$$:**
   $$
   3^{6} \cdot 3^{2} = 3^{6+2} = 3^{8}
   $$

3. **Calculate $$3^{8}$$ if desired:**
   $$
   3^{8} = 6561
   $$

**Final Answer:**
$$
3^{8} \quad \text{or} \quad 6561
$$

---

### **(e) $$\left(\frac{-12 x^{3}}{15 y^{-2}}\right)^{2}$$**

**Step-by-Step Simplification:**

1. **Simplify the expression inside the parentheses:**
   $$
   \frac{-12 x^{3}}{15 y^{-2}} = \frac{-12 x^{3} \cdot y^{2}}{15} = \frac{-4 x^{3} y^{2}}{5} \quad (\text{simplifying } -12/15 = -4/5)
   $$

2. **Apply the exponent to each factor:**
   $$
   \left(\frac{-4 x^{3} y^{2}}{5}\right)^{2} = \frac{(-4)^2 \cdot (x^{3})^2 \cdot (y^{2})^2}{5^2}
   $$

3. **Calculate each part:**
   - $$(-4)^2 = 16$$
   - $$(x^{3})^2 = x^{6}$$
   - $$(y^{2})^2 = y^{4}$$
   - $$5^2 = 25$$

4. **Combine the results:**
   $$
   \frac{16 x^{6} y^{4}}{25}
   $$

**Final Answer:**
$$
\frac{16 x^{6} y^{4}}{25}
$$

---

### **(f) $$\left(\frac{10 y^{4}}{4 y^{6}}\right)^{-2}$$**

**Step-by-Step Simplification:**

1. **Simplify the expression inside the parentheses:**
   $$
   \frac{10 y^{4}}{4 y^{6}} = \frac{10}{4} \cdot y^{4-6} = \frac{5}{2} \cdot y^{-2}
   $$

2. **Apply the exponent $$-2$$ to the entire expression:**
   $$
   \left(\frac{5}{2} \cdot y^{-2}\right)^{-2} = \left(\frac{5}{2}\right)^{-2} \cdot \left(y^{-2}\right)^{-2}
   $$

3. **Calculate each part:**
   - $$\left(\frac{5}{2}\right)^{-2} = \left(\frac{2}{5}\right)^{2} = \frac{4}{25}$$
   - $$\left(y^{-2}\right)^{-2} = y^{4}$$ (since $$-2 \times -2 = 4$$)

4. **Combine the results:**
   $$
   \frac{4 y^{4}}{25}
   $$

**Final Answer:**
$$
\frac{4 y^{4}}{25}
$$

---

**Summary of Simplified Expressions:**

- **(b)** $$-8 x^{9} y^{6}$$
- **(c)** $$3^{8}$$ or $$6561$$
- **(e)** $$\dfrac{16 x^{6} y^{4}}{25}$$
- **(f)** $$\dfrac{4 y^{4}}{25}$$
- **(b)** $$-8 x^{9} y^{6}$$ - **(c)** $$3^{8}$$ or $$6561$$ - **(e)** $$\dfrac{16 x^{6} y^{4}}{25}$$ - **(f)** $$\dfrac{4 y^{4}}{25}$$

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Algebra Questions