Question

Find the following product: \[\left( { - 4} \right) \times \left( { - 5} \right) \times \left( { - 8} \right) \times \left( { - 10} \right)\].

Answer 1

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

Complete step-by-step answer:
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 \[\left( { - 4} \right) \times \left( { - 5} \right) \times \left( { - 8} \right) \times \left( { - 10} \right)\] by multiplying two terms at once.
First, let us multiply \[ - 4\] by \[ - 5\].
The integer \[ - 4\] is the product of the numbers \[ - 1\] and 4.
Similarly, \[ - 5\] is the product of the numbers \[ - 1\] and 5.
Therefore, rewriting the expression \[\left( { - 4} \right) \times \left( { - 5} \right)\], we get
\[ \Rightarrow \left( { - 4} \right) \times \left( { - 5} \right) = \left( { - 1 \times 4} \right) \times \left( { - 1 \times 5} \right) \\
   \Rightarrow \left( { - 4} \right) \times \left( { - 5} \right) = \left( { - 1} \right) \times 4 \times \left( { - 1} \right) \times 5 \\\]
Rearranging the terms, we get
\[ \Rightarrow \left( { - 4} \right) \times \left( { - 5} \right) = \left( { - 1} \right) \times \left( { - 1} \right) \times 4 \times 5\]
The product of \[ - 1\] and \[ - 1\] is the square of \[ - 1\], that is 1.
Therefore, multiplying the terms in the expression, we get
\[ \Rightarrow \left( { - 4} \right) \times \left( { - 5} \right) = 1 \times 20 \\
   \Rightarrow \left( { - 4} \right) \times \left( { - 5} \right) = 20 \\\]
Now, let us multiply \[ - 8\] by \[ - 10\].
The integer \[ - 8\] is the product of the numbers \[ - 1\] and 8.
Similarly, \[ - 10\] is the product of the numbers \[ - 1\] and 10.
Therefore, rewriting the expression \[\left( { - 8} \right) \times \left( { - 10} \right)\], we get
\[\Rightarrow \left( { - 8} \right) \times \left( { - 10} \right) = \left( { - 1 \times 8} \right) \times \left( { - 1 \times 10} \right) \\
   \Rightarrow \left( { - 8} \right) \times \left( { - 10} \right) = \left( { - 1} \right) \times 8 \times \left( { - 1} \right) \times 10 \\\]
Rearranging the terms, we get
\[ \Rightarrow \left( { - 8} \right) \times \left( { - 10} \right) = \left( { - 1} \right) \times \left( { - 1} \right) \times 8 \times 10\]
The product of \[ - 1\] and \[ - 1\] is the square of \[ - 1\], that is 1.
Therefore, multiplying the terms in the expression, we get
\[\Rightarrow \left( { - 8} \right) \times \left( { - 10} \right) = 1 \times 80 \\
   \Rightarrow \left( { - 8} \right) \times \left( { - 10} \right) = 80 \\\]
Finally, we will find the value of the expression \[\left( { - 4} \right) \times \left( { - 5} \right) \times \left( { - 8} \right) \times \left( { - 10} \right)\].
Substituting \[\left( { - 4} \right) \times \left( { - 5} \right) = 20\] and \[\left( { - 8} \right) \times \left( { - 10} \right) = 80\] in the expression, we get
\[ \Rightarrow \left( { - 4} \right) \times \left( { - 5} \right) \times \left( { - 8} \right) \times \left( { - 10} \right) = 20 \times 80\]
Multiplying 20 by 80, we get
\[ \Rightarrow \left( { - 4} \right) \times \left( { - 5} \right) \times \left( { - 8} \right) \times \left( { - 10} \right) = 1600\]
\[\therefore \] We get the value of the product \[\left( { - 4} \right) \times \left( { - 5} \right) \times \left( { - 8} \right) \times \left( { - 10} \right)\] as 1600.

Note: We can also use exponents to simplify the given expression.
The integer \[ - 4\] is the product of the numbers \[ - 1\] and 4, \[ - 5\] is the product of the numbers \[ - 1\] and 5, \[ - 8\] is the product of the numbers \[ - 1\] and 8, and \[ - 10\] is the product of the numbers \[ - 1\] and 10.
Rewriting the given expression, we get
\[ \Rightarrow \left( { - 4} \right) \times \left( { - 5} \right) \times \left( { - 8} \right) \times \left( { - 10} \right) = \left( { - 1} \right) \times 4 \times \left( { - 1} \right) \times 5 \times \left( { - 1} \right) \times 8 \times \left( { - 1} \right) \times 10\]
Therefore, rewriting the expression using exponents, we get
\[ \Rightarrow \left( { - 4} \right) \times \left( { - 5} \right) \times \left( { - 8} \right) \times \left( { - 10} \right) = {\left( { - 1} \right)^4} \times 4 \times 5 \times 8 \times 10\]
We know that \[{\left( { - 1} \right)^n}\] is equal to 1 if \[n\] is an even natural number.
Therefore, \[{\left( { - 1} \right)^4} = 1\].
Simplifying the equation \[\left( { - 4} \right) \times \left( { - 5} \right) \times \left( { - 8} \right) \times \left( { - 10} \right) = {\left( { - 1} \right)^4} \times 4 \times 5 \times 8 \times 10\], we get
\[ \Rightarrow \left( { - 4} \right) \times \left( { - 5} \right) \times \left( { - 8} \right) \times \left( { - 10} \right) = 1 \times 4 \times 5 \times 8 \times 10\]
Multiplying the terms of the expression, we get
\[ \Rightarrow \left( { - 4} \right) \times \left( { - 5} \right) \times \left( { - 8} \right) \times \left( { - 10} \right) = 1600\]
\[\therefore \] We get the value of the product \[\left( { - 4} \right) \times \left( { - 5} \right) \times \left( { - 8} \right) \times \left( { - 10} \right)\] as 1600.