Question

What is Distributive Property?

Answer 1

Hint: In this question we will learn about distributive property for natural numbers, whole numbers, integers and rational numbers. Then we will learn about two types of distributive property that are left distributive property and right distributive property. Then we will do some examples related to them to understand this property properly.

Complete step-by-step answer:
In this question we have to explain about distributive property
Distributive Property.
As the name states this property distributes the operation.
So, basic property states that
$x\times \left( y+z \right)=(x\times y) + (x\times z) $
Where $x,y\And z$ can be natural numbers, whole numbers and integers.
Like this we can also define this property for subtraction operation.
$x\times \left( y-z \right)=(x\times y) - (x\times z) $
Where $x,y\And z$ can be natural numbers, whole numbers and integers.
Distributive property is of two types:
a) Left distributive property
b) Right distributive property
In Left distributive property, multiplication operation is on left side
$x\times \left( y+z \right)=(x\times y) + (x\times z) $
Like this we can also define this left distributive property for subtraction operation.
$x\times \left( y-z \right)=(x\times y) - (x\times z) $
Where $x,y\And z$ can be natural numbers, whole numbers and integers.
And
In Right distributive property, multiplication operation is on right side
$\left( y+z \right)\times x= (y\times x) + (z\times x) $
Like this we can also define this right distributive property for subtraction operation.
$\left( y-z \right)\times x=(y\times x) - (z\times x) $
Where $x,y\And z$ can be natural numbers, whole numbers and integers.

The distributive property does not apply to division in the same way as it does with multiplication, but the idea of distributing or “breaking apart” can be used in division.

Distributive Property for rational numbers are stated as
Left distributive property for rational numbers.
$\dfrac{a}{b}\times \left( \dfrac{c}{d}+\dfrac{e}{f} \right)=\dfrac{a}{b}\times \dfrac{c}{d}+\dfrac{a}{b}\times \dfrac{e}{f}$
Where $a,b,c,d,e,\And f$ are integers.
Provided $b,d,\And f$ should not be equal to $0$
$\begin{align}
  & b\ne 0 \\
 & d\ne 0 \\
 & f\ne 0 \\
\end{align}$
Now let us learn this distributive property from an example,
Example $1$ :
Solve
 $2\times \left( 5+8 \right)$ .
Now we will use left distributive property with multiplication operation
$\begin{align}
  & 2\times \left( 5+8 \right) \\
 & \Rightarrow 2\times 5+2\times 8 \\
 & \Rightarrow 10+16 \\
 & \Rightarrow 26 \\
\end{align}$
$\therefore 26$ is our required answer, $2\times \left( 5+8 \right)=26$

Example $2$ :
Solve
$5\times \left( 10-8 \right)$ .
Now we will use left distributive property with multiplication operation along with subtraction operation
$\begin{align}
  & 5\times \left( 10-8 \right) \\
 & \Rightarrow 5\times 10-5\times 8 \\
 & \Rightarrow 50-40 \\
 & \Rightarrow 10 \\
 & \\
\end{align}$
$\therefore 10$ is our required answer

Example $3$ :
Solve
 $\dfrac{1}{5}\times \left( \dfrac{5}{7}+\dfrac{10}{8} \right)$ .
Now we will use left distributive property of rational numbers with multiplication operation
$\begin{align}
  & \dfrac{1}{5}\times \left( \dfrac{5}{7}+\dfrac{10}{8} \right) \\
 & \Rightarrow \dfrac{1}{5}\times \dfrac{5}{7}+\dfrac{1}{5}\times \dfrac{10}{8} \\
 & \Rightarrow \dfrac{1}{7}+\dfrac{1}{4} \\
 & \Rightarrow \dfrac{4+7}{28} \\
\end{align}$
$\Rightarrow \dfrac{13}{28}$
$\therefore \dfrac{13}{28}$ is our required answer.

Note: In natural numbers, whole numbers, integers and rational numbers along with distributive property there are other properties also named as closure property, commutative property, additive identity and multiplicative identity. $0$ is called additive identity and $1$ is called multiplicative identity.