Answer :
To understand the inverse property of multiplication, we should analyze the given examples and then apply the property to fill in the blanks accurately.
The inverse property of multiplication states that for any real number \( a \), multiplying \( a \) by its inverse results in 1. The multiplicative inverse of a non-zero number \( a \) is \( \frac{1}{a} \).
Let's break down the examples provided:
1. [tex]\[ \frac{1}{5} \cdot 5 = 1 \][/tex]
Here, the number \( \frac{1}{5} \) is being multiplied by 5. Notice that \( \frac{1}{5} \) is the multiplicative inverse of 5 (since \( \frac{1}{5} = \frac{1}{a} \) when \( a = 5 \)), and thus their product is 1.
2. [tex]\[ \sqrt{2}\left(\frac{1}{\sqrt{2}}\right)=1 \][/tex]
In this example, \( \sqrt{2} \) is multiplied by \( \frac{1}{\sqrt{2}} \). Here, \( \frac{1}{\sqrt{2}} \) is the multiplicative inverse of \( \sqrt{2} \) (since \( \frac{1}{\sqrt{2}} = \frac{1}{a} \) when \( a = \sqrt{2} \)), and their product is 1.
Given these observations, we see that the inverse property of multiplication requires the product of a number and its multiplicative inverse to equal 1. However, this property does not hold for the number zero, because zero does not have a multiplicative inverse (since any number multiplied by zero is zero, not 1).
Therefore, the blanks can be filled in as follows:
Inverse property of multiplication: For all real numbers except \(\boxed{\text{zero}}\), \( a \cdot \boxed{\text{its multiplicative inverse}} = 1 \).
So the statement that best describes the inverse property of multiplication, with the blanks filled, is:
"For all real numbers except zero, [tex]\( a \cdot \)[/tex] its multiplicative inverse [tex]\( =1 \)[/tex]."
The inverse property of multiplication states that for any real number \( a \), multiplying \( a \) by its inverse results in 1. The multiplicative inverse of a non-zero number \( a \) is \( \frac{1}{a} \).
Let's break down the examples provided:
1. [tex]\[ \frac{1}{5} \cdot 5 = 1 \][/tex]
Here, the number \( \frac{1}{5} \) is being multiplied by 5. Notice that \( \frac{1}{5} \) is the multiplicative inverse of 5 (since \( \frac{1}{5} = \frac{1}{a} \) when \( a = 5 \)), and thus their product is 1.
2. [tex]\[ \sqrt{2}\left(\frac{1}{\sqrt{2}}\right)=1 \][/tex]
In this example, \( \sqrt{2} \) is multiplied by \( \frac{1}{\sqrt{2}} \). Here, \( \frac{1}{\sqrt{2}} \) is the multiplicative inverse of \( \sqrt{2} \) (since \( \frac{1}{\sqrt{2}} = \frac{1}{a} \) when \( a = \sqrt{2} \)), and their product is 1.
Given these observations, we see that the inverse property of multiplication requires the product of a number and its multiplicative inverse to equal 1. However, this property does not hold for the number zero, because zero does not have a multiplicative inverse (since any number multiplied by zero is zero, not 1).
Therefore, the blanks can be filled in as follows:
Inverse property of multiplication: For all real numbers except \(\boxed{\text{zero}}\), \( a \cdot \boxed{\text{its multiplicative inverse}} = 1 \).
So the statement that best describes the inverse property of multiplication, with the blanks filled, is:
"For all real numbers except zero, [tex]\( a \cdot \)[/tex] its multiplicative inverse [tex]\( =1 \)[/tex]."
