Question

Find the value of: $\left( { - 18} \right) \times \left( { - 5} \right) \times \left( { - 4} \right)$.

Answer 1

Hint: First, we will find the value of the product by multiplying two terms, and then multiplying the results with the third term to get the required value of the expression. The product of 2 negatives is always positive. The product of a negative number and a positive number is always negative.

Complete step-by-step solution:
Multiplication is the repeated addition of equal groups. It helps in adding multiple equal groups quickly. It is denoted by the symbol $ \times $. Brackets may also be used to denote multiplication.
Now, we will find the product $\left( { - 18} \right) \times \left( { - 5} \right) \times \left( { - 4} \right)$ by multiplying two terms at once.
First, let us multiply $ - 18$ by $ - 5$.
The integer $ - 18$ is the product of the numbers $ - 1$ and $18$.
The integer $ - 5$ is the product of the numbers $ - 1$ and $5$.
Therefore, rewriting the expression $ - 18 \times - 5$, we get
$ \Rightarrow \left( { - 18} \right) \times \left( { - 5} \right) = \left( { - 1 \times 18} \right) \times \left( { - 1 \times 5} \right)$
Open the brackets,
$ \Rightarrow \left( { - 18} \right) \times \left( { - 5} \right) = \left( { - 1} \right) \times 18 \times \left( { - 1} \right) \times 5$
Rearranging the terms, we get
$ \Rightarrow \left( { - 18} \right) \times \left( { - 5} \right) = \left( { - 1} \right) \times \left( { - 1} \right) \times 5 \times 18$
We know that the product of $ - 1$ and $ - 1$ is the square of $ - 1$, that is 1.
Therefore, by multiplying the terms in the expression, we get
$ \Rightarrow \left( { - 18} \right) \times \left( { - 5} \right) = 1 \times 90$
Simplify the term,
$ \Rightarrow \left( { - 18} \right) \times \left( { - 5} \right) = 90$
Now, let us multiply $90$ by $ - 4$.
The integer $ - 4$ is the product of the numbers $ - 1$ and $4$.
Therefore, rewriting the expression $90 \times - 4$, we get
$ \Rightarrow 90 \times \left( { - 4} \right) = 90 \times \left( { - 1 \times 4} \right)$
Open the brackets,
$ \Rightarrow 90 \times \left( { - 4} \right) = 90 \times \left( { - 1} \right) \times 4$
Rearranging the terms, we get
$ \Rightarrow 90 \times \left( { - 4} \right) = \left( { - 1} \right) \times 4 \times 90$
Therefore, by multiplying the terms in the expression, we get
$ \Rightarrow 90 \times \left( { - 4} \right) = \left( { - 1} \right) \times 360$
We know that the product of $ - 1$ with any number will give a negative of that number.
$\therefore 90 \times \left( { - 4} \right) = - 360$

Hence, the product of $\left( { - 18} \right) \times \left( { - 5} \right) \times \left( { - 4} \right)$ is $ - 360$.

Note: We can also use exponents to simplify the given expression.
The integer $ - 18$ is the product of the numbers $ - 1$ and $18$, $ - 5$ is the product of the numbers $ - 1$ and $5$ and $ - 4$ is the product of the numbers $ - 1$ and $4$.
Rewriting the given expression, we get
$ \Rightarrow \left( { - 18} \right) \times \left( { - 5} \right) \times \left( { - 4} \right) = \left( { - 1} \right) \times 18 \times \left( { - 1} \right) \times 5 \times \left( { - 1} \right) \times 4$
Therefore, by rewriting the expression using exponents, we get
$ \Rightarrow \left( { - 18} \right) \times \left( { - 5} \right) \times \left( { - 4} \right) = {\left( { - 1} \right)^3} \times 18 \times 5 \times 4$
We know that ${\left( { - 1} \right)^n}$ is equal to 1 if $n$ is an even natural number and is equal to $ - 1$ if $n$ is an odd natural number.
Simplifying the equation,
$ \Rightarrow \left( { - 18} \right) \times \left( { - 5} \right) \times \left( { - 4} \right) = \left( { - 1} \right) \times 18 \times 5 \times 4$
Multiplying the terms of the expression, we get
$\therefore \left( { - 18} \right) \times \left( { - 5} \right) \times \left( { - 4} \right) = - 360$
Hence, the product of $\left( { - 18} \right) \times \left( { - 5} \right) \times \left( { - 4} \right)$ is $ - 360$.