Question

Which property of multiplication is illustrated by:
$\dfrac{-4}{3}\times \left( \dfrac{-6}{5}+\dfrac{8}{7} \right)=\left( \dfrac{-4}{3}\times \dfrac{-6}{5} \right)+\left( \dfrac{-4}{3}\times \dfrac{8}{7} \right)$
A) Associative property
B) Commutative property
C) Distributive property
D) None of these

Answer 1

Hint:
Here we have to identify the property used in the given expression. According to the distributive property of multiplication, if $a$ , $b$ and $c$ are three numbers, then $a\times \left( b+c \right)=a\times b+a\times c$. We will observe that the given expression has been created by using the distributive property of multiplication.

Complete step by step solution:
The given expression is
  $\dfrac{-4}{3}\times \left( \dfrac{-6}{5}+\dfrac{8}{7} \right)=\left( \dfrac{-4}{3}\times \dfrac{-6}{5} \right)+\left( \dfrac{-4}{3}\times \dfrac{8}{7} \right)$
We know that the distributive property tells us how to solve the expression like $a\times \left( b+c \right)$. Using distributive property of multiplication, we will solve this expression by first distributing $a$ to $b$ and then distributing $a$ to $c$ i.e. we will multiply $a$ by $b$and $a$ by $c$, and then we will add these products to get the required solution of the expression.
Thus, the distributive property states that
For any real numbers; $a$ , $b$and $c$;
$a\times \left( b+c \right)=a\times b+a\times c$ ….. $\left( 1 \right)$
We will substitute the value of $a$ , $b$and $c$ as $\dfrac{-4}{3}$ , $\dfrac{-6}{5}$ and $\dfrac{8}{7}$ respectively in equation 1.
$\dfrac{-4}{3}\times \left( \dfrac{-6}{5}+\dfrac{8}{7} \right)=\dfrac{-4}{3}\times \dfrac{-6}{5}+\dfrac{-4}{3}\times \dfrac{8}{7}$
We can see that we have distributed $\dfrac{-4}{3}$ to $\dfrac{-6}{5}$ and then to $\dfrac{8}{7}$ i.e. we have multiplied $\dfrac{-4}{3}$ with $\dfrac{-6}{5}$ and then with $\dfrac{8}{7}$ and then we have added the two products.
Thus, we can see that the distributive property of multiplication has been used here.

Hence, the correct option is option C.

Note:
We generally use the distributive property when the two terms inside the parentheses can’t be added. When any one of the terms inside the bracket is a variable or they are not like terms, then we use the distributive property of multiplication. We have to apply the outside number to all of the terms inside the parentheses or brackets. In the distributive property of multiplication, multiplication is distributed over addition because the product of a term with a sum of terms is the same as the sum of all of the products of the terms.