\( 1 \leftarrow \) Find each of the following products. Write the answers in simplest form. \( \begin{array}{ll}\text { a. } \frac{33}{77} \cdot \frac{33}{132} & \text { b. } \frac{a}{v} \cdot \frac{v^{2}}{a^{2}} \quad \text { c. } \frac{q x}{t} \cdot \frac{t^{2} n}{q^{3} x^{2}} \\ \text { a. } \frac{33}{77} \cdot \frac{33}{132}=\square \text { (Type an integer or a simplified fraction.) }\end{array} \)

To find the products, let's simplify each one step by step. **a.** For \( \frac{33}{77} \cdot \frac{33}{132} \): First, simplify the fractions individually. - \( \frac{33}{77} = \frac{33 \div 11}{77 \div 11} = \frac{3}{7} \) - \( \frac{33}{132} = \frac{33 \div 33}{132 \div 33} = \frac{1}{4} \) Now, multiply the simplified fractions: \[ \frac{3}{7} \cdot \frac{1}{4} = \frac{3 \cdot 1}{7 \cdot 4} = \frac{3}{28} \] So, \( \frac{33}{77} \cdot \frac{33}{132} = \frac{3}{28} \). **b.** For \( \frac{a}{v} \cdot \frac{v^{2}}{a^{2}} \): Multiply the fractions directly: \[ \frac{a}{v} \cdot \frac{v^{2}}{a^{2}} = \frac{a \cdot v^2}{v \cdot a^2} = \frac{a \cdot v^{2}}{a^{2} \cdot v} \] Now, simplify by canceling \( a \) and \( v \): \[ = \frac{v}{a} \] So, \( \frac{a}{v} \cdot \frac{v^{2}}{a^{2}} = \frac{v}{a} \). **c.** For \( \frac{qx}{t} \cdot \frac{t^{2}n}{q^{3}x^{2}} \): Multiply the fractions directly: \[ \frac{qx}{t} \cdot \frac{t^{2}n}{q^{3}x^{2}} = \frac{qx \cdot t^{2}n}{t \cdot q^{3}x^{2}} = \frac{q \cdot x \cdot t^{2}n}{t \cdot q^{3}x^{2}} \] Now, simplify by canceling: 1. One \( t \) from \( t^{2} \) gives \( t \). 2. One \( q \) gives \( \frac{1}{q^{2}} \) as \( q^{3} \) becomes \( q^{2} \). 3. One \( x \) gives \( \frac{1}{x} \) as \( x^{2} \) becomes \( x \). Putting it all together: \[ = \frac{tn}{q^{2}x} \] So, \( \frac{qx}{t} \cdot \frac{t^{2}n}{q^{3}x^{2}} = \frac{tn}{q^{2}x} \). Overall, here’s your summary: a. \( \frac{33}{77} \cdot \frac{33}{132} = \frac{3}{28} \) b. \( \frac{a}{v} \cdot \frac{v^{2}}{a^{2}} = \frac{v}{a} \) c. \( \frac{qx}{t} \cdot \frac{t^{2}n}{q^{3}x^{2}} = \frac{tn}{q^{2}x} \)