Simplify: 1. $$ 49 a^{2} imes \frac{2 a b}{7} \div 8 a^{2} b $$ 3. $$ 4(-2 p q)(3 p q) $$ 5. $$ 2 p q \div 3 p^{2} q imes 4 p q^{3} $$ 7. $$ \left(\frac{8 a^{2} b c^{2}}{4 a b^{2} c} ight)^{2} $$ 9. $$ \left(\frac{54 x^{6} y}{2 x^{3} y^{4}} ight)^{\frac{1}{3}} $$

Question

Simplify: 1. $$ 49 a^{2} 	imes \frac{2 a b}{7} \div 8 a^{2} b $$ 3. $$ 4(-2 p q)(3 p q) $$ 5. $$ 2 p q \div 3 p^{2} q 	imes 4 p q^{3} $$ 7. $$ \left(\frac{8 a^{2} b c^{2}}{4 a b^{2} c}ight)^{2} $$ 9. $$ \left(\frac{54 x^{6} y}{2 x^{3} y^{4}}ight)^{\frac{1}{3}} $$

UpStudy AI · Accepted Answer

Sure, let's simplify each expression step by step.

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**1. $$ 49 a^{2} \times \frac{2 a b}{7} \div 8 a^{2} b $$**

**Step-by-Step Simplification:**
1. Multiply $$49 a^{2}$$ by $$\frac{2 a b}{7}$$:
   $$
   49 a^{2} \times \frac{2 a b}{7} = \frac{49 \times 2}{7} a^{2} \times a \times b = 14 a^{3} b
   $$
2. Divide the result by $$8 a^{2} b$$:
   $$
   \frac{14 a^{3} b}{8 a^{2} b} = \frac{14}{8} \times a^{3-2} \times b^{1-1} = \frac{7}{4} a
   $$

**Simplified Form:**
$$
\frac{7}{4} a
$$

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**3. $$ 4(-2 p q)(3 p q) $$**

**Simplification:**
$$
4 \times (-2) \times 3 \times p \times p \times q \times q = -24 p^{2} q^{2}
$$

**Simplified Form:**
$$
-24 p^{2} q^{2}
$$

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**5. $$ 2 p q \div 3 p^{2} q \times 4 p q^{3} $$**

**Step-by-Step Simplification:**
1. Perform the division:
   $$
   \frac{2 p q}{3 p^{2} q} = \frac{2}{3} \times \frac{1}{p} = \frac{2}{3 p}
   $$
2. Multiply by $$4 p q^{3}$$:
   $$
   \frac{2}{3 p} \times 4 p q^{3} = \frac{8}{3} q^{3}
   $$

**Simplified Form:**
$$
\frac{8}{3} q^{3}
$$

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**7. $$ \left(\frac{8 a^{2} b c^{2}}{4 a b^{2} c}\right)^{2} $$**

**Step-by-Step Simplification:**
1. Simplify inside the parentheses:
   $$
   \frac{8 a^{2} b c^{2}}{4 a b^{2} c} = 2 \times a^{2-1} \times b^{1-2} \times c^{2-1} = \frac{2 a c}{b}
   $$
2. Square the result:
   $$
   \left(\frac{2 a c}{b}\right)^{2} = \frac{4 a^{2} c^{2}}{b^{2}}
   $$

**Simplified Form:**
$$
\frac{4 a^{2} c^{2}}{b^{2}}
$$

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**9. $$ \left(\frac{54 x^{6} y}{2 x^{3} y^{4}}\right)^{\frac{1}{3}} $$**

**Step-by-Step Simplification:**
1. Simplify inside the parentheses:
   $$
   \frac{54 x^{6} y}{2 x^{3} y^{4}} = 27 \times x^{6-3} \times y^{1-4} = 27 x^{3} y^{-3} = \frac{27 x^{3}}{y^{3}}
   $$
2. Take the cube root:
   $$
   \left(\frac{27 x^{3}}{y^{3}}\right)^{\frac{1}{3}} = \frac{3 x}{y}
   $$

**Simplified Form:**
$$
\frac{3 x}{y}
$$

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**Summary of Simplified Expressions:**

1. $$ \frac{7}{4} a $$
3. $$ -24 p^{2} q^{2} $$
5. $$ \frac{8}{3} q^{3} $$
7. $$ \frac{4 a^{2} c^{2}}{b^{2}} $$
9. $$ \frac{3 x}{y} $$
1. $$ \frac{7}{4} a $$ 3. $$ -24 p^{2} q^{2} $$ 5. $$ \frac{8}{3} q^{3} $$ 7. $$ \frac{4 a^{2} c^{2}}{b^{2}} $$ 9. $$ \frac{3 x}{y} $$

Simplify: 1. \( 49 a^{2} \times \frac{2 a b}{7} \div 8 a^{2} b \) 3. \( 4(-2 p q)(3 p q) \) 5. \( 2 p q \div 3 p^{2} q \times 4 p q^{3} \) 7. \( \left(\frac{8 a^{2} b c^{2}}{4 a b^{2} c}\right)^{2} \) 9. \( \left(\frac{54 x^{6} y}{2 x^{3} y^{4}}\right)^{\frac{1}{3}} \)

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