Chapter DPT, Problem 1E

Replace each question mark with an appropriate expression that will illustrate the use of the indicated real number property:

$\begin{array}{l}\left(\text{A}\right)\text{ Commutative (}\cdot \text{)}:x(y+z)=?\\ \left(\text{B}\right)\text{ Associative }(+):2+\text{(}x+y)=?\\ \left(\text{C}\right)\text{ Distributive: (2}+\text{3})x=?\end{array}$ :

To determine

An appropriate expression that may replace each question mark, and illustrate the use of the indicated real number property:

(A) Commutative $(\cdot )$ : $x\left(y+z\right)=?$

(B) Associative $(+)$ : $2+\left(x+y\right)=?$

(C) Distributive: $\left(2+3\right)x=?$

Answer to Problem 1E

(A) The required expression for the Commutative property of multiplication is $x\left(y+z\right)=\left(y+z\right)x$.

(B) The required expression for the Associative property of addition is $2+\left(x+y\right)=\left(2+x\right)+y$.

(C) The required expression for the Distributive property is $\left(2+3\right)x=2x+3x$.

Explanation of Solution

(A)

Consider the given expression,

$x\left(y+z\right)=?$

Now, commutative property of multiplication states that $\forall a,b\in ℝ$, $ab=ba$.

Here, take $x$ as $a$ and $\left(y+z\right)$ as $b$.

Then, by the commutative property,

$x\left(y+z\right)=\left(y+z\right)x$

Hence, the required expression for the Commutative property of multiplication is $x\left(y+z\right)=\left(y+z\right)x$.

(B)

Consider the given expression,

$2+\left(x+y\right)=?$

Now, associative property of addition states that $\forall a,b,c\in ℝ$, $a+\left(b+c\right)=\left(a+b\right)+c$.

Here, take $2$ as $a$ and $\left(x+y\right)$ as $\left(b+c\right)$.

Then, by the associative property,

$2+\left(x+y\right)=\left(2+x\right)+y$

Hence, the required expression for the Associative property of addition is $2+\left(x+y\right)=\left(2+x\right)+y$.

(C)

Consider the given expression,

$\left(2+3\right)x=?$

Now, distributive property states that for every $\forall a,b,c\in ℝ$, $\left(a+b\right)c=ac+bc$.

Here, take $\left(2+3\right)$ as $\left(a+b\right)$ and $x$ as $c$,

Then, by the distributive property,

$\left(2+3\right)x=2x+3x$

Hence, the required expression for the Distributive property is $\left(2+3\right)x=2x+3x$.