Chapter DPT, Problem 1E

Work all of the problems in this self-test without using a calculator. Then check your work by consulting the answers in the back of the book. Where weaknesses show up, use the reference that follows each answer to find the section in the test that provides the necessary review.

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

(A)    Commutative (·): x(y + z) = ?

(B)    Associative (+): 2 + (x + y) = ?

(C)    Distributive: (2 + 3)x = ?

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

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

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 $2x+3x$.