Question

Explain how the commutative and associative properties of multiplication can help you evaluate the product $5 · 17 · 2$ mentally

Short answer

$\text{5} · 17 · 2 = 170$

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 – Explain how the commutative and associative properties of multiplication can help you evaluate the product 5 x 17 x 2 mentally.

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

Associative property of Multiplication

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

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

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

Commutative Property of Multiplication:

Property: localid="1647518898866" $\text{a} · \text{b} =\text{b} · \text{a}$or $\text{a} · \text{b} · \text{c}=\text{b} · \text{c} · \text{a}$$\text{a} · \text{b} · \text{c}=\text{b} · \text{c} · \text{a}$

If you multiply two or three real numbers in any order, their product will always be the same.

Example: $\text{12} · \text{4} =\text{4} · \text{12} = 48$or $\text{5} · \text{6} · \text{4}=\text{6} · \text{4} · \text{5} \text{=} \text{120}$

Step-3 – Explain how the commutative and associative properties of multiplication can help you evaluate the product 5 x 17 x 2 mentally.

Commutative Property of Multiplication states that if you multiply two or three real numbers in any order, their product will always be the same.

Property: $\text{a} · \text{b} =\text{b} · \text{a}$or $\text{a} · \text{b} · \text{c}=\text{b} · \text{c} · \text{a}$

Example: $\text{12} · \text{4} =\text{4} · \text{12} = 48$or $\text{5} · \text{6} · \text{4}=\text{6} · \text{4} · \text{5} \text{=} \text{120}$

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

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

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

In short we can write:

Commutative Property of Multiplication: $\text{a} · \text{b} =\text{b} · \text{a}$or $\text{a} · \text{b} · \text{c}=\text{b} · \text{c} · \text{a}$

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

So, we can multiply the numbers in any order.

localid="1649347774162" $\begin{array}{l}\text{5}·17·2=170\\ =5(17·2)(\text{byAssociativeProperty)}\\ =5(2·17)\left(\text{byCommutativeProperty}\right)\\ =5·34=170\end{array}$

$\text{5} · 17 · 2 = 170$