Let's verify the associative property under multiplication for the given rational numbers. The associative property of multiplication states that for any three numbers [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex], the equation [tex]\( x \times (y \times z) = (x \times y) \times z \)[/tex] should hold true.
Given:
[tex]\[ x = \frac{1}{2}, \quad y = \frac{-2}{3}, \quad z = \frac{3}{4} \][/tex]
### Step 1: Calculate [tex]\( y \times z \)[/tex]
First, we need to find the product of [tex]\( y \)[/tex] and [tex]\( z \)[/tex]:
[tex]\[ y \times z = \frac{-2}{3} \times \frac{3}{4} \][/tex]
Multiplying the numerators together and the denominators together:
[tex]\[ y \times z = \frac{(-2) \times 3}{3 \times 4} = \frac{-6}{12} \][/tex]
Simplify the fraction:
[tex]\[ y \times z = \frac{-6}{12} = \frac{-1}{2} \][/tex]
### Step 2: Calculate [tex]\( x \times (y \times z) \)[/tex]
Now, we calculate [tex]\( x \times (y \times z) \)[/tex]:
[tex]\[ x \times (y \times z) = \frac{1}{2} \times \frac{-1}{2} \][/tex]
Multiplying the numerators together and the denominators together:
[tex]\[ x \times (y \times z) = \frac{1 \times -1}{2 \times 2} = \frac{-1}{4} \][/tex]
### Step 3: Calculate [tex]\( x \times y \)[/tex]
Next, we find the product of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x \times y = \frac{1}{2} \times \frac{-2}{3} \][/tex]
Multiplying the numerators together and the denominators together:
[tex]\[ x \times y = \frac{1 \times -2}{2 \times 3} = \frac{-2}{6} \][/tex]
Simplify the fraction:
[tex]\[ x \times y = \frac{-2}{6} = \frac{-1}{3} \][/tex]
### Step 4: Calculate [tex]\( (x \times y) \times z \)[/tex]
Finally, we calculate [tex]\( (x \times y) \times z \)[/tex]:
[tex]\[ (x \times y) \times z = \frac{-1}{3} \times \frac{3}{4} \][/tex]
Multiplying the numerators together and the denominators together:
[tex]\[ (x \times y) \times z = \frac{-1 \times 3}{3 \times 4} = \frac{-3}{12} \][/tex]
Simplify the fraction:
[tex]\[ (x \times y) \times z = \frac{-3}{12} = \frac{-1}{4} \][/tex]
### Step 5: Compare the Results
We have found:
[tex]\[ x \times (y \times z) = \frac{-1}{4} \][/tex]
[tex]\[ (x \times y) \times z = \frac{-1}{4} \][/tex]
Both sides of the equation are equal, confirming that:
[tex]\[ x \times (y \times z) = (x \times y) \times z \][/tex]
Thus, the associative property under multiplication holds true for the given values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex].
