Question

Name the property under multiplication used in each of the following.
(i)\[\dfrac{{ - 4}}{5} \times 1 = 1 \times \dfrac{{ - 4}}{5} = \dfrac{{ - 4}}{5}\]
(ii)\[\dfrac{{ - 13}}{{17}} \times \dfrac{{ - 2}}{7} = \dfrac{{ - 2}}{7} \times \dfrac{{ - 13}}{{17}}\]
(iii)\[\dfrac{{ - 19}}{{29}} \times \dfrac{{29}}{{ - 19}} = 1\]

Answer 1

Hint: According to the given question, we will separately solve all the parts by calculating a and b from the given equations. Make one of the identities from the five different properties under the multiplication. After solving the name that property is used.

Complete step-by-step answer:
As, we are given the different parts and we have to name the property under multiplication which is most applicable from the five different properties of multiplication that are Associative property, commutative property, distributive property, multiplicative identity and multiplicative inverse. Using the formulas we will proceed for :
\[\dfrac{{ - 4}}{5} \times 1 = 1 \times \dfrac{{ - 4}}{5} = \dfrac{{ - 4}}{5}\]
Here, we will suppose \[a = \dfrac{{ - 4}}{5}\] , \[b = 1\]
So substituting the values into variables we get,
\[a \times b = b \times a = a\]
Also from the above equation it is seen that here b is 1. So, we substitute b=1 to make it identity,
So, it is in the form of
\[a \times 1 = 1 \times a = a\]
Hence, it is a multiplicative identity 1 under multiplication.
\[\dfrac{{ - 13}}{{17}} \times \dfrac{{ - 2}}{7} = \dfrac{{ - 2}}{7} \times \dfrac{{ - 13}}{{17}}\]
Here, we will suppose \[a = \dfrac{{ - 13}}{{17}}\] , \[b = \dfrac{{ - 2}}{7}\]
So substituting the values into variables we get,
\[a \times b = b \times a\]
Also in above equation it is clearly seen that it is in the form of:
\[a \times b = b \times a\]
Hence, it is a commutative property under multiplication.
$\Rightarrow$ \[\dfrac{{ - 19}}{{29}} \times \dfrac{{29}}{{ - 19}} = 1\]
Here, we will suppose \[a = - 19\] , \[b = 29\]
So substituting the values into variables we get,
$\Rightarrow$ \[\dfrac{a}{b} \times \dfrac{b}{a} = 1\]
Also in above equation it is clearly seen that it is in the form of:
$\Rightarrow$ \[\dfrac{a}{b} \times \dfrac{b}{a} = 1\]
Hence, it is a multiplicative inverse property under multiplication.

Note: To solve these types of questions, we need to remember all the five different properties under multiplication and what does it mean in the form of the equation. Using these properties, questions can be of different properties just we have to follow the same method as shown above.