The following examples illustrate the inverse property of multiplication. Study the examples, then choose the statement that best describes the property.

[tex]\[
\begin{array}{l}
\frac{1}{5} \cdot 5=1 \\
\sqrt{2}\left(\frac{1}{\sqrt{2}}\right)=1
\end{array}
\][/tex]

Inverse property of multiplication: For all real numbers except [tex]\(0\)[/tex],
[tex]\[ a \cdot \frac{1}{a} = 1 \][/tex]



Answer :

To understand the inverse property of multiplication, let’s go through the given examples and then deduce the property.

### Examples
1. [tex]\(\frac{1}{5} \cdot 5 = 1\)[/tex]
- Here, [tex]\(\frac{1}{5}\)[/tex] is the multiplicative inverse of 5.

2. [tex]\(\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 1\)[/tex]
- Here, [tex]\(\frac{1}{\sqrt{2}}\)[/tex] is the multiplicative inverse of [tex]\(\sqrt{2}\)[/tex].

### Explanation
The inverse property of multiplication can be stated as:
> For each real number [tex]\(a\)[/tex], there exists another real number [tex]\(b\)[/tex] such that:
[tex]\[ a \cdot b = 1 \][/tex]
where [tex]\(b = \frac{1}{a}\)[/tex], which is known as the "multiplicative inverse" or "reciprocal" of [tex]\(a\)[/tex].

This property holds for all real numbers except one specific number. Which number is it?

#### Zero as a Special Case
- The number [tex]\(0\)[/tex] does not have a multiplicative inverse. This is because there is no number that can be multiplied by [tex]\(0\)[/tex] to result in [tex]\(1\)[/tex]. Mathematically, [tex]\(\frac{1}{0}\)[/tex] is undefined.

### Conclusion
The inverse property of multiplication holds for all real numbers except [tex]\(0\)[/tex], because the multiplicative inverse of zero is not defined.

### Final Answer
The inverse property of multiplication: For all real numbers except [tex]\[\boxed{0}\][/tex] [tex]\[\boxed{a \cdot \frac{1}{a} = 1}\][/tex]