Question

Question: Evaluate the expression. Justify each of your steps.

$-4\left(9\right)(-5)$

Short answer

The value of the expression is180.

Step-by-step solution

Step-1 – What are real numbers? 

Real numbers are simply the combination of Whole numbers, Integers, Rational and Irrational numbers in the number system.

Step-2 – Associative Property of Multiplication of Real Numbers: 

Suppose a, b, and c is three real numbers.

Associative property of Multiplication

Property:$\text{(a\hspace{0.17em}\hspace{0.17em}}·\text{\hspace{0.17em}\hspace{0.17em}b)\hspace{0.17em}\hspace{0.17em}}·\text{\hspace{0.17em}\hspace{0.17em}c}=\text{a\hspace{0.17em}\hspace{0.17em}}·(\text{\hspace{0.17em}b\hspace{0.17em}\hspace{0.17em}}·\text{\hspace{0.17em}\hspace{0.17em}c\hspace{0.17em}})$

If you are multiplying three real numbers, the product is always the same regardless of their grouping.

Example:$\text{(6\hspace{0.17em}\hspace{0.17em}}·\text{\hspace{0.17em}\hspace{0.17em}4)\hspace{0.17em}\hspace{0.17em}}·\text{\hspace{0.17em}\hspace{0.17em}3}=\text{6\hspace{0.17em}\hspace{0.17em}}·(\text{\hspace{0.17em}4\hspace{0.17em}\hspace{0.17em}}·\text{\hspace{0.17em}\hspace{0.17em}3\hspace{0.17em}})=72$

$\text{(a\hspace{0.17em}\hspace{0.17em}}·\text{\hspace{0.17em}\hspace{0.17em}b)\hspace{0.17em}\hspace{0.17em}}·\text{\hspace{0.17em}\hspace{0.17em}c}=\text{a\hspace{0.17em}\hspace{0.17em}}·(\text{\hspace{0.17em}b\hspace{0.17em}\hspace{0.17em}}·\text{\hspace{0.17em}\hspace{0.17em}c\hspace{0.17em}})$

Step-3 – Evaluate the expression – 4 (9) (– 5) and justify each of your steps.

Associative property of Multiplication states thatif you are multiplying three real numbers, the product is always the same regardless of their grouping.

Property:$\text{(a\hspace{0.17em}\hspace{0.17em}}·\text{\hspace{0.17em}\hspace{0.17em}b)\hspace{0.17em}\hspace{0.17em}}·\text{\hspace{0.17em}\hspace{0.17em}c}=\text{a\hspace{0.17em}\hspace{0.17em}}·(\text{\hspace{0.17em}b\hspace{0.17em}\hspace{0.17em}}·\text{\hspace{0.17em}\hspace{0.17em}c\hspace{0.17em}})$

Example:$\text{(6\hspace{0.17em}\hspace{0.17em}}·\text{\hspace{0.17em}\hspace{0.17em}4)\hspace{0.17em}\hspace{0.17em}}·\text{\hspace{0.17em}\hspace{0.17em}3}=\text{6\hspace{0.17em}\hspace{0.17em}}·(\text{\hspace{0.17em}4\hspace{0.17em}\hspace{0.17em}}·\text{\hspace{0.17em}\hspace{0.17em}3\hspace{0.17em}})=72$

Associative Property of Multiplication: $\text{(a\hspace{0.17em}\hspace{0.17em}}·\text{\hspace{0.17em}\hspace{0.17em}b)\hspace{0.17em}\hspace{0.17em}}·\text{\hspace{0.17em}\hspace{0.17em}c}=\text{a\hspace{0.17em}\hspace{0.17em}}·(\text{\hspace{0.17em}b\hspace{0.17em}\hspace{0.17em}}·\text{\hspace{0.17em}\hspace{0.17em}c\hspace{0.17em}})$

So, we can multiply the numbers in any order.

$\begin{array}{l}-4\left(9\right)(-5)\\ (-4·9)(-5)(\text{Associative\hspace{0.17em}\hspace{0.17em}Property\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}Multiplication)}\\ -4(9·-5)(\text{Associative\hspace{0.17em}\hspace{0.17em}Property\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}Multiplication)}\\ -4·-45=180\end{array}$

Thus, the value of the expression is 180.