Question

Find each of the following products: $( - 3) \times ( - 3) \times ( - 3) \times \ldots 6$ times.

Answer 1

Hint: Here we have to find the products of integers. Integers can be defined as the set of all-natural numbers, zero, and the negatives of natural numbers but without fractional components. Here, we will use the multiplication rule of integers which states that while multiplying any two integers with each other, first find the products of the integers without considering the signs and after that see the signs of numbers.

Complete step by step answer:
In the given question we have to find the products of integers. Integers can be defined as numbers that can be either zero, positive or negative numbers but integers can never be a fraction, a decimal or a percent. The symbol used to represent integers is Z. Examples of integers are $ - 2, - 1,0,1 \ldots $
So, we will use the multiplication rule of integers which states that while multiplying any two integers with each other, first find the products of the integers without considering the signs and after that see the signs of numbers. If the sign of both the numbers is the same then the product is positive whereas if the sign of both the numbers is different then the product is negative.
We have $( - 3) \times ( - 3) \times ( - 3) \times ( - 3) \times ( - 3) \times ( - 3)$
We can write the above equation as
$ \Rightarrow \left\{ {( - 3 \times - 3) \times ( - 3 \times - 3) \times ( - 3 \times - 3)} \right\}$
Here the sign of both the numbers is same so the product of the number is positive.
$ \Rightarrow \left\{ {9 \times 9 \times 9} \right\}$
On multiplying the numbers. We get
$ \Rightarrow 81 \times 9$
$ \Rightarrow 729$
We can also write the product of the number in the power form. So,
$ \Rightarrow {( - 3)^6} = 729$
Hence, the product of $( - 3) \times ( - 3) \times ( - 3) \times \ldots 6$ times is equal to $729$.

Note:
Note that integers follow the closure property under the operations of addition, subtraction, and multiplication. This means that if we add, subtract or multiply two given integers the result we obtained is also an integer. But integers are not closed under division, since the integers need to be in fraction form which is not the definition of integers.