Chapter DPT, Problem 1E

(A)

To determine

To find: The equivalent expression of $x\left(y+z\right)$ by using the commutative property.

Answer to Problem 1E

The expression $x\left(y+z\right)$ is equivalent to $\left(y+z\right)x$.

Explanation of Solution

Given:

The expression is $x\left(y+z\right)$.

Property used:

Commutative property:

“The order of factors does not affect the result of expression”.

That is, if x commutes with y under the operation $"\cdot "$, then $x\cdot y=y\cdot x$.

Illustration:

The factors of the given expression $x\left(y+z\right)$ are $x$ and $\left(y+z\right)$.

Apply the commutative property by changing the order of factors to obtain the required equivalent expression.

That is, $x\left(y+z\right)$ is altered to $\left(y+z\right)x$.

Therefore, the expression $x\left(y+z\right)$ is equivalent to $\left(y+z\right)x$.

(B)

To determine

To find: The equivalent expression of $2+\left(x+y\right)$ by using the associative property.

Answer to Problem 1E

The expression $2+\left(x+y\right)$ is equivalent to $\left(2+x\right)+y$.

Explanation of Solution

Given:

The expression is $2+\left(x+y\right)$.

Property used:

Associative property:

The grouping of any two numbers in an expression which contains more than two terms does affect the resulting expression.

That is, the property of associative under “$+$” operation states that $\left(a+b\right)+c=a+\left(b+c\right)$.

Illustration:

Apply the associative property by changing the group of terms to obtain the required equivalent expression.

That is, $2+\left(x+y\right)$ is altered to $\left(2+x\right)+y$.

Therefore, the expression of $2+\left(x+y\right)$ is equivalent to $\left(2+x\right)+y$.

(C)

To determine

To find: The equivalent expression of $\left(2+3\right)x$ by using the distributive property.

Answer to Problem 1E

The expression $\left(2+3\right)x$ is equivalent to $2x+3x$.

Explanation of Solution

Given:

The expression is $\left(2+3\right)x$.

Property used:

Distributive property:

The property of left distributive under multiplication over addition state that $\left(a+b\right)c=ac+bc$.

Illustration:

Apply the left distributive property under multiplication over addition to obtain the required equivalent expression.

That is, $\left(2+3\right)x$ is altered to $2x+3x$.

Hence, the expression of $\left(2+3\right)x$ equivalent to $2x+3x$.