### Video Transcript

Which of the following properties
of multiplication is represented in the equation five-sevenths times eight-thirds
multiplied by two-ninths is equal to five-sevenths multiplied by eight-thirds times
two-ninths? Option (A) associative
property. Option (B) multiplicative identity
property. Option (C) commutative
property. Option (D) distributive
property. Or is it option (E) multiplicative
inverse property?

In this question, we are given an
equation. And we need to determine which of
five given properties of the multiplication of rational numbers is represented in
the equation. To do this, we can begin by looking
at the equation. We see that we have the products of
three rational numbers. We can also see that we are
evaluating the products in a different order on each side of the equation.

We can then recall that the
property that allows us to evaluate the products of rational numbers in any order is
called the associative property of the multiplication of rational numbers. More formally, it tells us that for
any rational numbers 𝑎, 𝑏, and 𝑐, we have that 𝑎 times 𝑏 multiplied by 𝑐 is
equal to 𝑎 multiplied by 𝑏 times 𝑐. We can see that this is the same as
the given equation with 𝑎 equal to five-sevenths, 𝑏 equal to eight-thirds, and 𝑐
equal to two-ninths. For due diligence, we can also
recall what each of the other four properties of multiplication of rational numbers
tells us.

First, we recall that the
multiplicative identity property tells us that multiplication by one leaves any
rational number unchanged. So, if 𝑎 is a rational number,
then 𝑎 times one is equal to 𝑎.

Second, we can recall that the
commutativity property of the multiplication of rational numbers tells us that we
can reorder the product of any rational numbers. More formally, if 𝑎 and 𝑏 are
rational numbers, then 𝑎 times 𝑏 is the same as 𝑏 times 𝑎.

Third, we can recall that the
distributive property of the multiplication of rational numbers over addition tells
us that we can distribute multiplication over addition by multiplying every term by
the factor. More formally, if 𝑎, 𝑏, and 𝑐
are rational numbers, then 𝑎 multiplied by 𝑏 plus 𝑐 is equal to 𝑎 times 𝑏 plus
𝑎 times 𝑐.

Finally, we can recall that the
multiplicative inverse property of rational numbers tells us that all nonzero
rational numbers have a multiplicative inverse. In general, the multiplicative
inverse of 𝑎 over 𝑏 will be 𝑏 over 𝑎. So their product is one, provided
𝑎 and 𝑏 are nonzero integers.

However, only the associative
property of multiplication is shown in the given equation. So the answer is option (A), the
associative property.