Certainly! Let's verify if the equation [tex]\( x \times (y \times z) = (x \times y) \times z \)[/tex] holds true for the given values [tex]\( x = \frac{1}{2} \)[/tex], [tex]\( y = -\frac{2}{3} \)[/tex], and [tex]\( z = \frac{1}{4} \)[/tex].
### Step-by-Step Solution
1. Calculate [tex]\( y \times z \)[/tex]:
[tex]\[
y \times z = \left(-\frac{2}{3}\right) \times \left(\frac{1}{4}\right)
\][/tex]
Multiplying the fractions, we get:
[tex]\[
y \times z = \frac{-2 \times 1}{3 \times 4} = \frac{-2}{12} = -\frac{1}{6}
\][/tex]
2. Calculate [tex]\( x \times (y \times z) \)[/tex]:
Now that we have [tex]\( y \times z = -\frac{1}{6} \)[/tex], we can calculate:
[tex]\[
x \times (y \times z) = \frac{1}{2} \times \left(-\frac{1}{6}\right)
\][/tex]
Multiplying the fractions, we get:
[tex]\[
x \times (y \times z) = \frac{1 \times -1}{2 \times 6} = \frac{-1}{12} = -\frac{1}{12}
\][/tex]
3. Calculate [tex]\( x \times y \)[/tex]:
[tex]\[
x \times y = \left(\frac{1}{2}\right) \times \left(-\frac{2}{3}\right)
\][/tex]
Multiplying the fractions, we get:
[tex]\[
x \times y = \frac{1 \times -2}{2 \times 3} = \frac{-2}{6} = -\frac{1}{3}
\][/tex]
4. Calculate [tex]\( (x \times y) \times z \)[/tex]:
Now that we have [tex]\( x \times y = -\frac{1}{3} \)[/tex], we can calculate:
[tex]\[
(x \times y) \times z = \left(-\frac{1}{3}\right) \times \frac{1}{4}
\][/tex]
Multiplying the fractions, we get:
[tex]\[
(x \times y) \times z = \frac{-1 \times 1}{3 \times 4} = \frac{-1}{12} = -\frac{1}{12}
\][/tex]
5. Comparison:
We can now compare the two results:
[tex]\[
x \times (y \times z) = -\frac{1}{12}
\][/tex]
[tex]\[
(x \times y) \times z = -\frac{1}{12}
\][/tex]
Since the results are equal, we have verified that:
[tex]\[
x \times (y \times z) = (x \times y) \times z
\][/tex]
Therefore, the equality [tex]\( x \times (y \times z) = (x \times y) \times z \)[/tex] holds true for the given values [tex]\( x = \frac{1}{2} \)[/tex], [tex]\( y = -\frac{2}{3} \)[/tex], and [tex]\( z = \frac{1}{4} \)[/tex].
