8. Write $$ \frac{3 x^{2}+6 x}{2 x y+4 y} imes \frac{2 y}{9 x^{2}} $$ in its simplified form and substitute the value to evaluate, given that $$ x=1 $$ and $$ y=2 $$

Question

8. Write $$ \frac{3 x^{2}+6 x}{2 x y+4 y} 	imes \frac{2 y}{9 x^{2}} $$ in its simplified form and substitute the value to evaluate, given that $$ x=1 $$ and $$ y=2 $$

UpStudy AI · Accepted Answer

To simplify and evaluate the expression:

$$
\frac{3x^{2} + 6x}{2xy + 4y} \times \frac{2y}{9x^{2}}
$$

given $$ x = 1 $$ and $$ y = 2 $$, follow these steps:

### **Step 1: Factor and Simplify the Expression**

1. **Factor the Numerator and Denominator:**

$$
   3x^{2} + 6x = 3x(x + 2)
   $$
   $$
   2xy + 4y = 2y(x + 2)
   $$

2. **Rewrite the Expression with Factored Terms:**

$$
   \frac{3x(x + 2)}{2y(x + 2)} \times \frac{2y}{9x^{2}}
   $$

3. **Cancel Common Terms $$(x + 2)$$:**

$$
   \frac{3x}{2y} \times \frac{2y}{9x^{2}} 
   $$

4. **Multiply the Fractions:**

$$
   \frac{3x \cdot 2y}{2y \cdot 9x^{2}} = \frac{6xy}{18x^{2}y}
   $$

5. **Simplify the Fraction:**

$$
   \frac{6xy}{18x^{2}y} = \frac{6}{18} \times \frac{xy}{x^{2}y} = \frac{1}{3x}
   $$

So, the simplified form of the expression is:

$$
   \frac{1}{3x}
   $$

### **Step 2: Substitute the Given Values $$ x = 1 $$ and $$ y = 2 $$**

$$
\frac{1}{3x} = \frac{1}{3 \times 1} = \frac{1}{3}
$$

### **Final Answer**

$$
\frac{1}{3}
$$
The simplified form of the expression is $$ \frac{1}{3x} $$. Substituting $$ x = 1 $$ gives $$ \frac{1}{3} $$.

Write in its simplified
form and substitute the value to evaluate,
given that and

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