Chapter 7.2, Problem 51IP

To determine

Simplify and name of the property used in the statement $24\cdot \left(-15\right)+36\cdot 15$ .

Answer to Problem 51IP

  $180$

Distributive property

Explanation of Solution

Given information:

The statement, $24\cdot \left(-15\right)+36\cdot 15$

Concept Used:

Commutative property:

• $a+b=b+a$
• $a\cdot b=b\cdot a$

Associative property:

• $\left(a+b\right)+c=a+\left(b+c\right)$
• $\left(a\cdot b\right)\cdot c=a\cdot \left(b\cdot c\right)$

Distributive property:

• $a\left(b+c\right)=ab+ac$
• $\left(b+c\right)a=ba+ca$

Additive identity:

• $a+0=0+a=a$

Multiplicative identity:

• $a\cdot 1=1\cdot a=a$

Zero property of Multiplication:

• $a\cdot 0=0\cdot a=0$

Calculation:

Here in the given statement, observe that sequence of the addition of first two numbers is change in the right side. It is of the form $a\left(b+c\right)=ab+ac$ . So, the property shown by the given statement is distributive property of addition as below:

  $\begin{array}{c}24\cdot (-15)+36\cdot 15\\ =15(-24+36)\\ \text{ (distributive property)}\\ =15(12)\end{array}$

Simplify $15(12)$ as below

  $\begin{array}{c}15(12)\\ =180\end{array}$

Thus, the given statement shows the distributive property.