We'll verify two key properties of multiplication:
1. Commutative Property: This property states that for any numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex], the result of [tex]\(a \times b\)[/tex] is the same as [tex]\(b \times a\)[/tex].
2. Associative Property: This property states that for any numbers [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], the result of [tex]\(a \times (b \times c)\)[/tex] is the same as [tex]\((a \times b) \times c\)[/tex].
Let's verify these steps with specific values:
### 1. Verifying the Commutative Property:
Given:
[tex]\[ a = \frac{3}{4} \][/tex]
[tex]\[ b = \frac{-1}{2} \][/tex]
Calculate [tex]\( a \times b \)[/tex]:
[tex]\[ a \times b = \left( \frac{3}{4} \right) \times \left( \frac{-1}{2} \right) = -\frac{3}{8} \][/tex]
Now, calculate [tex]\( b \times a \)[/tex]:
[tex]\[ b \times a = \left( \frac{-1}{2} \right) \times \left( \frac{3}{4} \right) = -\frac{3}{8} \][/tex]
Since both products yield the same result [tex]\(-\frac{3}{8}\)[/tex], the commutative property holds true:
[tex]\[ a \times b = b \times a = -0.375 \][/tex]
### 2. Verifying the Associative Property:
Given:
[tex]\[ a = \frac{3}{4} \][/tex]
[tex]\[ b = \frac{-1}{2} \][/tex]
[tex]\[ c = \frac{-7}{5} \][/tex]
First, calculate [tex]\( b \times c \)[/tex]:
[tex]\[ b \times c = \left( \frac{-1}{2} \right) \times \left( \frac{-7}{5} \right) = \frac{7}{10} \][/tex]
Next, calculate [tex]\( a \times (b \times c) \)[/tex]:
[tex]\[ a \times (b \times c) = \left( \frac{3}{4} \right) \times \left( \frac{7}{10} \right) = \frac{21}{40} = 0.525 \][/tex]
Now, calculate [tex]\( a \times b \)[/tex]:
[tex]\[ a \times b = \left( \frac{3}{4} \right) \times \left( \frac{-1}{2} \right) = -\frac{3}{8} \][/tex]
Then, calculate [tex]\((a \times b) \times c\)[/tex]:
[tex]\[ (a \times b) \times c = \left( -\frac{3}{8} \right) \times \left( \frac{-7}{5} \right) = \frac{21}{40} = 0.525 \][/tex]
Since both products yield the same result [tex]\(0.525\)[/tex], the associative property holds true:
[tex]\[ a \times (b \times c) = (a \times b) \times c = 0.525 \][/tex]
In conclusion, the given values satisfy both the commutative property [tex]\(a \times b = b \times a = -0.375\)[/tex] and the associative property [tex]\(a \times (b \times c) = (a \times b) \times c = 0.525\)[/tex].
