The inverse property of multiplication states that for every non-zero real number [tex]\(a\)[/tex], there exists a number [tex]\(b\)[/tex], specifically [tex]\(b = \frac{1}{a}\)[/tex], such that the product of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] is equal to 1.
Let's analyze the given examples:
1. [tex]\(\frac{1}{5} \cdot 5 = 1\)[/tex]
- Here, [tex]\(a = 5\)[/tex] and [tex]\(b = \frac{1}{5}\)[/tex].
- According to the inverse property, [tex]\(5 \cdot \frac{1}{5} = 1\)[/tex], which is true.
2. [tex]\(\sqrt{2} \left( \frac{1}{\sqrt{2}} \right) = 1\)[/tex]
- Here, [tex]\(a = \sqrt{2}\)[/tex] and [tex]\(b = \frac{1}{\sqrt{2}}\)[/tex].
- According to the inverse property, [tex]\(\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 1\)[/tex], which is true.
Thus, these examples illustrate that for any number [tex]\(a\)[/tex], there is a corresponding number [tex]\(\frac{1}{a}\)[/tex] that satisfies the condition [tex]\(a \cdot \frac{1}{a} = 1\)[/tex].
However, it's important to note that this property cannot hold for [tex]\(a = 0\)[/tex] because division by zero is undefined.
Therefore, the correct statement describing the inverse property of multiplication should be:
For all real numbers except [tex]\(0\)[/tex], [tex]\(a \cdot \frac{1}{a} = 1\)[/tex].
The missing value in the given property statement is [tex]\(0\)[/tex]. So, it should read:
"For all real numbers except [tex]\(0\)[/tex], [tex]\(a \cdot \frac{1}{a} = 1\)[/tex]."
