(231500 Khawaja Quiz\# 01 asic Mathematics Sections -G Time: 30 minute Quia Marks \# $$ \# 01 $$ simplify the radical; $$ \sqrt{18 x^{4} y^{3}} $$ $$ \# 02 $$ Simplify the exponent; $$ \left(\frac{6 m n^{-2}}{3 m^{-1} n^{2}}\right)^{-3} $$ \# 03 Solve the Adding Polynomial $$ \begin{array}{l} \text { Solve the Adding } \ \left(x^{4}-3 x^{3}+x^{2}\right),\left(-x^{3}-2 x^{2}+3 x\right),\left(3 x^{2}-5\right) \end{array} $$

Question

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Simplify the expression by following steps:
- step0: Solution:
  $$\sqrt{18x^{4}y^{3}}$$
- step1: Write as a product:
  $$\sqrt{9\times 2x^{4}y^{3}}$$
- step2: Write in exponential form:
  $$\sqrt{3^{2}\times 2x^{4}y^{3}}$$
- step3: Rewrite the exponent:
  $$\sqrt{3^{2}\times 2x^{4}y^{2+1}}$$
- step4: Expand the expression:
  $$\sqrt{3^{2}\times 2x^{4}y^{2}\times y}$$
- step5: Reorder the terms:
  $$\sqrt{3^{2}x^{4}y^{2}\times 2y}$$
- step6: Use the properties of radicals:
  $$\sqrt{3^{2}x^{4}y^{2}}\times \sqrt{2y}$$
- step7: Simplify the root:
  $$3yx^{2}\sqrt{2y}$$
Calculate or simplify the expression $$ (6 * m * n^(-2) / (3 * m^(-1) * n^2))^(-3) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(\frac{6mn^{-2}}{\left(3m^{-1}n^{2}\right)}\right)^{-3}$$
- step1: Remove the parentheses:
  $$\left(\frac{6mn^{-2}}{3m^{-1}n^{2}}\right)^{-3}$$
- step2: Multiply by $$a^{-n}:$$
  $$\left(\frac{6mn^{-2}mn^{-2}}{3}\right)^{-3}$$
- step3: Reduce the fraction:
  $$\left(\frac{2m^{2}}{n^{4}}\right)^{-3}$$
- step4: Transform the expression:
  $$\left(\frac{n^{4}}{2m^{2}}\right)^{3}$$
- step5: Rewrite the expression:
  $$\frac{\left(n^{4}\right)^{3}}{\left(2m^{2}\right)^{3}}$$
- step6: Evaluate the power:
  $$\frac{n^{12}}{8m^{6}}$$
Calculate or simplify the expression $$ (x^4 - 3 * x^3 + x^2) + (-x^3 - 2 * x^2 + 3 * x) + (3 * x^2 - 5) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(x^{4}-3x^{3}+x^{2}\right)+\left(-x^{3}-2x^{2}+3x\right)+\left(3x^{2}-5\right)$$
- step1: Remove the parentheses:
  $$x^{4}-3x^{3}+x^{2}-x^{3}-2x^{2}+3x+3x^{2}-5$$
- step2: Subtract the terms:
  $$x^{4}-4x^{3}+2x^{2}+3x-5$$
Let's solve each part of the quiz step by step.

### $$ \# 01 $$ Simplify the radical: $$ \sqrt{18 x^{4} y^{3}} $$

1. **Factor the expression inside the square root:**
   $$
   18 = 9 \times 2 = 3^2 \times 2
   $$
   Thus, we can rewrite the expression as:
   $$
   \sqrt{18 x^{4} y^{3}} = \sqrt{3^2 \times 2 \times x^{4} \times y^{3}}
   $$

2. **Apply the square root to each factor:**
   $$
   = \sqrt{3^2} \times \sqrt{2} \times \sqrt{x^{4}} \times \sqrt{y^{3}} = 3 \times \sqrt{2} \times x^{2} \times \sqrt{y^{3}}
   $$

3. **Simplify $$ \sqrt{y^{3}} $$:**
   $$
   \sqrt{y^{3}} = y^{1} \times \sqrt{y} = y\sqrt{y}
   $$

4. **Combine everything:**
   $$
   = 3yx^{2}\sqrt{2y}
   $$

**Final Answer for $$ \# 01 $$:** $$ 3yx^{2}\sqrt{2y} $$

---

### $$ \# 02 $$ Simplify the exponent: $$ \left(\frac{6 m n^{-2}}{3 m^{-1} n^{2}}\right)^{-3} $$

1. **Simplify the fraction inside the parentheses:**
   $$
   \frac{6 m n^{-2}}{3 m^{-1} n^{2}} = \frac{6}{3} \cdot \frac{m}{m^{-1}} \cdot \frac{n^{-2}}{n^{2}} = 2 \cdot m^{2} \cdot n^{-4}
   $$

2. **Now raise the simplified expression to the power of -3:**
   $$
   (2 m^{2} n^{-4})^{-3} = 2^{-3} \cdot (m^{2})^{-3} \cdot (n^{-4})^{-3} = \frac{1}{8} \cdot m^{-6} \cdot n^{12}
   $$

3. **Rewrite the expression:**
   $$
   = \frac{n^{12}}{8 m^{6}}
   $$

**Final Answer for $$ \# 02 $$:** $$ \frac{n^{12}}{8 m^{6}} $$

---

### $$ \# 03 $$ Solve the Adding Polynomial

We need to add the polynomials:
$$
\left(x^{4}-3 x^{3}+x^{2}\right) + \left(-x^{3}-2 x^{2}+3 x\right) + \left(3 x^{2}-5\right)
$$

1. **Combine like terms:**
   - $$ x^{4} $$ terms: $$ x^{4} $$
   - $$ x^{3} $$ terms: $$ -3x^{3} - x^{3} = -4x^{3} $$
   - $$ x^{2} $$ terms: $$ x^{2} - 2x^{2} + 3x^{2} = 2x^{2} $$
   - $$ x $$ terms: $$ 3x $$
   - Constant terms: $$ -5 $$

2. **Write the final polynomial:**
   $$
   x^{4} - 4x^{3} + 2x^{2} + 3x - 5
   $$

**Final Answer for $$ \# 03 $$:** $$ x^{4} - 4x^{3} + 2x^{2} + 3x - 5 $$

---

### Summary of Answers:
- $$ \# 01 $$: $$ 3yx^{2}\sqrt{2y} $$
- $$ \# 02 $$: $$ \frac{n^{12}}{8 m^{6}} $$
- $$ \# 03 $$: $$ x^{4} - 4x^{3} + 2x^{2} + 3x - 5 $$
$$ \# 01 $$: $$ 3yx^{2}\sqrt{2y} $$ $$ \# 02 $$: $$ \frac{n^{12}}{8 m^{6}} $$ $$ \# 03 $$: $$ x^{4} - 4x^{3} + 2x^{2} + 3x - 5 $$

(231500 Khawaja
Quiz# 01
asic Mathematics Sections -G Time: 30 minute
Quia Marks #
simplify the radical;
Simplify the exponent;
# 03 Solve the Adding Polynomial

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(231500 Khawaja Quiz# 01 asic Mathematics Sections -G Time: 30 minute Quia Marks # simplify the radical; Simplify the exponent; # 03 Solve the Adding Polynomial