Chapter 1.4, Problem 4STP

For Exercises $1-5,$ choose the correct letter.

Which expression is equivalent to $-a\cdot b$?

F. $a\cdot (-b)$

G. $b-a$

H. $(-a)(-b)$

I. $-a+b$

To determine

To choose: The expression that is equivalent to $-a\cdot b$ .

Answer to Problem 4STP

The first option F $a\left(-b\right)$ is correct.

Given:

The expressions are $a\left(-b\right),b-a,\left(-a\right)\left(-b\right)\text{ and }\left(-a\right)\left(-b\right)$ .

Explanation:

The first expression F $a\left(-b\right)$ can be solved as follows:

  $\begin{array}{c}a\cdot \left(-b\right)=a\left(-1\right)\left(b\right)\\ =\left(-1\right)\left(a\right)\left(b\right)\text{ commutative property }\\ =-a\cdot b\left[\because \left(-a\right)=\left(-1\right)a\right]\end{array}$

The second expression $b-a$ can be solved as follows:

  $b-a=b-\left(1\right)a$

It is the difference of two numbers but the given expression is the product.

The third expression $\left(-a\right)\left(-b\right)$ can be solved as follows:

  $\begin{array}{c}\left(-a\right)\left(-b\right)=-1\left(a\right)\cdot \left(-1\right)b\left[\because -x=\left(-1\right)x\right]\\ =-1\left(-1\right)a\cdot b\left[\text{ commutative property}\right]\\ =ab\left[\left(-1\right)\left(-1\right)=1\right]\end{array}$

The fourth expression $-a+b$ can be solved as follows:

  $-a+b=-1\left(a\right)+b$

It is the difference of two numbers but the given expression is the product.

Therefore, it can be seen from all the answers that the first option F $a\left(-b\right)$ is correct.

Explanation of Solution