Exercise 10 Page 73

Practice makes perfect

Consider the given equation.

a * b ? = b * a To decide if it is true or false for all real numbers a and b we can use the Properties of Real Numbers. The Commutative Property of Multiplication tells us that changing the order of the factors does not change the product. Let's try it here!

a * b? =b * a

CommutativePropMult

Commutative Property of Multiplication

a * b=a * b

Using the Commutative Property of Multiplication, we can see that this statement is true.

This time, we have been given the following equation.

a( b * c) ? = ab * acWe are now going to use the Properties of Real Numbers to decide if the statement is true. The Associative Property of Multiplication tells us that the grouping of factors does not affect the product. Therefore, we can remove the parentheses. Next, we are going to simplify further.

a( b * c)? =ab * ac

AssociativePropMult

Associative Property of Multiplication

a * b * c ? = ab * ac

CommutativePropMult

Commutative Property of Multiplication

a * b * c ? = a* a * b* c

ProdToPowTwoFac

a* a=a^2

a * b * c ≠ a^2 * b* c

Having simplified both sides, we see that the expressions are different since the exponents of a are different. Therefore, the expressions are not equivalent and the statement is false. To find a counterexample, we will substitute arbitrary values for a, b, and c. Let's use a= 2, b= 3, and c= 5.

a( b * c)? =ab * ac

SubstituteValues

Substitute values

2( 3 * 5)? = 2( 3) * 2( 5)

Multiply

30≠60

Since 30≠ 60 we know that the given expressions are not equivalent. Finding just one counterexample is enough to show non-equivalence. In fact, these expressions will be equivalent only for a=1.

Exercise 10 Page 73

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