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Evaluate the following using suitable identity: \[{\left( {998} \right)^3}\].

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Last updated date: 13th Jun 2024
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Answer
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Hint: Here, we will expand the given expression in the form of an algebraic identity. Then by using the suitable algebraic identity we will simplify the expression and by using the BODMAS rule, we will find the value of the algebraic expression.

Formula used:
The cube of the difference of two numbers is given by an algebraic identity \[{\left( {a - b} \right)^3} = {a^3} - 3{a^2}b + 3a{b^2} - {b^3}\].

Complete Step by Step Solution:
We are given an arithmetic expression \[{\left( {998} \right)^3}\].
Now, we will rewrite the given expression, which will be in the form of an Algebraic Identity, we get
\[{\left( {998} \right)^3} = {\left( {1000 - 2} \right)^3}\]
Thus, the expanded algebraic expression is in the form of an algebraic identity as \[{\left( {a - b} \right)^3}\] .
Now, by using an algebraic identity \[{\left( {a - b} \right)^3} = {a^3} - 3{a^2}b + 3a{b^2} - {b^3}\], we get
\[ \Rightarrow {\left( {1000 - 2} \right)^3} = {\left( {1000} \right)^3} - 3{\left( {1000} \right)^2}\left( 2 \right) + 3\left( {1000} \right){\left( 2 \right)^2} - {\left( 2 \right)^3}\]
Applying the exponent on the terms, we get
\[ \Rightarrow {\left( {1000 - 2} \right)^3} = 1000000000 - 3\left( {1000000} \right)\left( 2 \right) + 3\left( {1000} \right)\left( 4 \right) - \left( 8 \right)\]
Now, by multiplying the terms in the expression, we get
\[ \Rightarrow {\left( {1000 - 2} \right)^3} = 1000000000 - 6000000 + 12000 - 8\]
Now, by simplifying the expression using the BODMAS rule, we get
\[ \Rightarrow {\left( {1000 - 2} \right)^3} = 1000012000 - 6000008\]
Subtracting the terms, we get
\[ \Rightarrow {\left( {1000 - 2} \right)^3} = 994011992\]
\[ \Rightarrow {\left( {998} \right)^3} = 994011992\]

Therefore, the value of \[{\left( {998} \right)^3}\]is \[994011992\].

Note:
 We know that a binomial expression is defined as an algebraic expression having two terms and these terms must be unlike. We know that an equality relation which is true for all the values of the variables is called an Algebraic Identity. Here, we have used BODMAS rule which states that the first operation has to be done which is in the brackets, next the operation applies on the indices or order, then it moves on to the division and multiplication and then using addition and subtraction we will simplify the expression. If addition or subtraction and division or multiplication is in the same calculations, then it has to be done from left to right.