Ouestion 5 Expand and simplify as far as possible: 5.1. $$ 0.2 x y\left(0.1 x^{2}-0.3 x y\right) $$ (3) 5.2. $$ 2-\frac{y-4}{4}-\frac{1}{8}(y+1) $$ 5.3. $$ \left(\frac{-4 a^{-1} b}{-12 a^{-2} b^{-3}}\right)^{2} $$ 5.4. $$ \frac{2^{x} \cdot 4^{x+2}}{8^{x+3}} $$ 5.5. $$ \frac{6 a^{2} b-9 a b^{2}}{12 a^{3} b^{2}-18 a^{2} b^{3}} $$ 5.6. $$ \left(2 p^{2}-3\right)(p-5)-(p+3)^{2} $$

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Simplify the expression by following steps:
- step0: Subtract the terms:
  $$\left(2p^{2}-3\right)\left(p-5\right)-\left(p+3\right)^{2}$$
- step1: Expand the expression:
  $$2p^{3}-10p^{2}-3p+15-\left(p+3\right)^{2}$$
- step2: Expand the expression:
  $$2p^{3}-10p^{2}-3p+15-p^{2}-6p-9$$
- step3: Subtract the terms:
  $$2p^{3}-11p^{2}-9p+6$$
Calculate or simplify the expression $$ (2^{x} * 4^{x+2})/(8^{x+3}) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(2^{x}\times 4^{x+2}\right)}{8^{x+3}}$$
- step1: Remove the parentheses:
  $$\frac{2^{x}\times 4^{x+2}}{8^{x+3}}$$
- step2: Multiply:
  $$\frac{2^{3x+4}}{8^{x+3}}$$
- step3: Factor the expression:
  $$\frac{2^{3x+4}}{2^{3x+9}}$$
- step4: Divide the numbers:
  $$\frac{1}{2^{3x+9-\left(3x+4\right)}}$$
- step5: Subtract the terms:
  $$\frac{1}{2^{5}}$$
- step6: Evaluate the power:
  $$\frac{1}{32}$$
Expand the expression $$ ((-4 a^{-1} b)/(-12 a^{-2} b^{-3}))^{2 $$
Simplify the expression by following steps:
- step0: Evaluate the power:
  $$\left(\frac{\left(-4a^{-1}b\right)}{\left(-12a^{-2}b^{-3}\right)}\right)^{2}$$
- step1: Remove the parentheses:
  $$\left(\frac{-4a^{-1}b}{-12a^{-2}b^{-3}}\right)^{2}$$
- step2: Divide the terms:
  $$\left(\frac{ab^{4}}{3}\right)^{2}$$
- step3: Evaluate the power:
  $$\frac{a^{2}b^{8}}{9}$$
Expand the expression $$ 0.2 x y(0.1 x^{2}-0.3 x y) $$
Simplify the expression by following steps:
- step0: Simplify:
  $$0.2xy\left(0.1x^{2}-0.3xy\right)$$
- step1: Rewrite the expression:
  $$0.2\left(0.1x^{2}-0.3xy\right)xy$$
- step2: Multiply the expression:
  $$0.2\left(0.1xy-0.3y^{2}\right)x^{2}$$
- step3: Rearrange the terms:
  $$0.02x^{3}y-0.06y^{2}x^{2}$$
Calculate or simplify the expression $$ (6 a^{2} b-9 a b^{2})/(12 a^{3} b^{2}-18 a^{2} b^{3}) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(6a^{2}b-9ab^{2}\right)}{\left(12a^{3}b^{2}-18a^{2}b^{3}\right)}$$
- step1: Remove the parentheses:
  $$\frac{6a^{2}b-9ab^{2}}{12a^{3}b^{2}-18a^{2}b^{3}}$$
- step2: Rewrite the expression:
  $$\frac{6a^{2}b-9b^{2}a}{2ab\left(6a^{2}b-9b^{2}a\right)}$$
- step3: Reduce the fraction:
  $$\frac{1}{2ab}$$
Calculate or simplify the expression $$ 2-(y-4)/4-(1/8)(y+1) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$2-\frac{\left(y-4\right)}{4}-\frac{1}{8}\left(y+1\right)$$
- step1: Remove the parentheses:
  $$2-\frac{y-4}{4}-\frac{1}{8}\left(y+1\right)$$
- step2: Multiply the terms:
  $$2-\frac{y-4}{4}-\frac{1}{8}y-\frac{1}{8}$$
- step3: Rewrite the expression:
  $$2-\frac{y-4}{4}-\frac{y}{8}-\frac{1}{8}$$
- step4: Reduce fractions to a common denominator:
  $$\frac{2\times 4\times 2}{4\times 2}-\frac{\left(y-4\right)\times 2}{4\times 2}-\frac{y}{8}-\frac{1}{8}$$
- step5: Multiply the numbers:
  $$\frac{2\times 4\times 2}{8}-\frac{\left(y-4\right)\times 2}{4\times 2}-\frac{y}{8}-\frac{1}{8}$$
- step6: Multiply the numbers:
  $$\frac{2\times 4\times 2}{8}-\frac{\left(y-4\right)\times 2}{8}-\frac{y}{8}-\frac{1}{8}$$
- step7: Transform the expression:
  $$\frac{2\times 4\times 2-\left(y-4\right)\times 2-y-1}{8}$$
- step8: Multiply the terms:
  $$\frac{16-\left(y-4\right)\times 2-y-1}{8}$$
- step9: Multiply the terms:
  $$\frac{16-\left(2y-8\right)-y-1}{8}$$
- step10: Subtract the terms:
  $$\frac{23-3y}{8}$$
Let's go through each part of the problem step by step.

### 5.1 Expand and simplify:
$$
0.2 x y(0.1 x^{2}-0.3 x y)
$$
**Step 1:** Distribute $$0.2xy$$ to both terms inside the parentheses:
$$
0.2xy \cdot 0.1x^2 - 0.2xy \cdot 0.3xy
$$
**Step 2:** Perform the multiplication:
$$
0.02x^3y - 0.06y^2x^2
$$
**Final Result:**
$$
0.02x^3y - 0.06y^2x^2
$$

### 5.2 Expand and simplify:
$$
2 - \frac{y-4}{4} - \frac{1}{8}(y+1)
$$
**Step 1:** Simplify each term:
- The first term is $$2$$.
- The second term becomes $$-\frac{y}{4} + 1$$.
- The third term becomes $$-\frac{y}{8} - \frac{1}{8}$$.

**Step 2:** Combine all terms:
$$
2 + 1 - \frac{y}{4} - \frac{y}{8} - \frac{1}{8}
$$
**Step 3:** Convert $$2$$ to eighths:
$$
\frac{16}{8} + \frac{8}{8} - \frac{y}{4} - \frac{y}{8} - \frac{1}{8} = \frac{23 - 3y}{8}
$$
**Final Result:**
$$
\frac{23 - 3y}{8}
$$

### 5.3 Expand and simplify:
$$
\left(\frac{-4 a^{-1} b}{-12 a^{-2} b^{-3}}\right)^{2}
$$
**Step 1:** Simplify the fraction:
$$
\frac{-4 a^{-1} b}{-12 a^{-2} b^{-3}} = \frac{4}{12} \cdot \frac{b}{a^{-1} b^{-3}} = \frac{1}{3} \cdot a^{1} b^{4} = \frac{a b^{4}}{3}
$$
**Step 2:** Square the result:
$$
\left(\frac{a b^{4}}{3}\right)^{2} = \frac{a^{2} b^{8}}{9}
$$
**Final Result:**
$$
\frac{a^{2} b^{8}}{9}
$$

### 5.4 Expand and simplify:
$$
\frac{2^{x} \cdot 4^{x+2}}{8^{x+3}}
$$
**Step 1:** Rewrite $$4$$ and $$8$$ in terms of $$2$$:
$$
4^{x+2} = (2^2)^{x+2} = 2^{2(x+2)} = 2^{2x + 4}
$$
$$
8^{x+3} = (2^3)^{x+3} = 2^{3(x+3)} = 2^{3x + 9}
$$
**Step 2:** Substitute back into the expression:
$$
\frac{2^{x} \cdot 2^{2x + 4}}{2^{3x + 9}} = \frac{2^{3x + 4}}{2^{3x + 9}} = 2^{(3x + 4) - (3x + 9)} = 2^{-5}
$$
**Final Result:**
$$
\frac{1}{32} \quad \text{or} \quad 0.03125
$$

### 5.5 Expand and simplify:
$$
\frac{6 a^{2} b - 9 a b^{2}}{12 a^{3} b^{2} - 18 a^{2} b^{3}}
$$
**Step 1:** Factor out common terms:
Numerator: $$3ab(2a - 3b)$$

Denominator: $$6a^{2}b^{2}(2a - 3b)$$

**Step 2:** Cancel the common factor:
$$
\frac{3ab(2a - 3b)}{6a^{2}b^{2}(2a - 3b)} = \frac{3}{6ab} = \frac{1}{2ab}
$$
**Final Result:**
$$
\frac{1}{2ab}
$$

### 5.6 Expand and simplify:
$$
\left(2 p^{2}-3\right)(p-5)-(p+3)^{2}
$$
**Step 1:** Expand both parts:
$$
(2p^{2} - 3)(p - 5) = 2p^{3} - 10p^{2} - 3p + 15
$$
$$
(p + 3)^{2} = p^{2} + 6p + 9
$$
**Step 2:** Combine:
$$
2p^{3} - 10p^{2} - 3p + 15 - (p^{2} + 6p + 9)
$$
**Step 3:** Distribute the negative:
$$
2p^{3} - 10p^{2} - 3p + 15 - p^{2} - 6p - 9
$$
**Step 4:** Combine like terms:
$$
2p^{3} - 11p^{2} - 9p + 6
$$
**Final Result:**
$$
2p^{3} - 11p^{2} - 9p + 6
$$

### Summary of Results:
- **5.1:** $$0.02x^3y - 0.06y^2x^2$$
- **5.2:** $$\frac{23 - 3y}{8}$$
- **5.3:** $$\frac{a^{2} b^{8}}{9}$$
- **5.4:** $$\frac{1}{32}$$ or $$0.03125$$
- **5.5:** $$\frac{1}{2ab}$$
- **5.6:** $$2p^{3} - 11p^{2} - 9p + 6$$
- **5.1:** $$0.02x^3y - 0.06y^2x^2$$ - **5.2:** $$\frac{23 - 3y}{8}$$ - **5.3:** $$\frac{a^{2} b^{8}}{9}$$ - **5.4:** $$\frac{1}{32}$$ or $$0.03125$$ - **5.5:** $$\frac{1}{2ab}$$ - **5.6:** $$2p^{3} - 11p^{2} - 9p + 6$$

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