Question

Compute the value of \[{\left( {3m - 2n} \right)^3}\].

Answer 1

Hint: Here, we need to compute the given expression. We will use the algebraic identity for the cube of the difference of two numbers. Then, we will simplify the expression to find the required value of the given expression.

Formula Used:
 We will use the following formulas:
1.The cube of the difference of two numbers is given by the algebraic identity \[{\left( {a - b} \right)^3} = {a^3} - {b^3} - 3ab\left( {a - b} \right)\].
2.An expression of the form \[{\left( {ab} \right)^n}\] can be written as \[{a^n}{b^n}\].

Complete step-by-step answer:
We will use the algebraic identity for the cube of the difference of two numbers to expand the given expression, and then simplify it to obtain the required value.
Substituting \[a = 3m\] and \[b = 2n\] in the identity \[{\left( {a - b} \right)^3} = {a^3} - {b^3} - 3ab\left( {a - b} \right)\], we get
\[ \Rightarrow {\left( {3m - 2n} \right)^3} = {\left( {3m} \right)^3} - {\left( {2n} \right)^3} - 3\left( {3m} \right)\left( {2n} \right)\left( {3m - 2n} \right)\]
We know that an expression of the form \[{\left( {ab} \right)^n}\] can be written as \[{a^n}{b^n}\].
Therefore, we can rewrite \[{\left( {3m} \right)^3}\] as \[{3^3}{m^3}\].
Similarly, we can rewrite \[{\left( {2n} \right)^3}\] as \[{2^3}{n^3}\].
Substituting \[{\left( {3m} \right)^3} = {3^3}{m^3}\] and \[{\left( {2n} \right)^3} = {2^3}{n^3}\] in the equation \[{\left( {3m - 2n} \right)^3} = {\left( {3m} \right)^3} - {\left( {2n} \right)^3} - 3\left( {3m} \right)\left( {2n} \right)\left( {3m - 2n} \right)\], we get
\[ \Rightarrow {\left( {3m - 2n} \right)^3} = {3^3}{m^3} - {2^3}{n^3} - 3\left( {3m} \right)\left( {2n} \right)\left( {3m - 2n} \right)\]
Applying the exponents on the bases, we get
\[ \Rightarrow {\left( {3m - 2n} \right)^3} = 27{m^3} - 8{n^3} - 3\left( {3m} \right)\left( {2n} \right)\left( {3m - 2n} \right)\]
Multiplying the terms in the expression, we get
\[ \Rightarrow {\left( {3m - 2n} \right)^3} = 27{m^3} - 8{n^3} - 18mn\left( {3m - 2n} \right)\]
Multiplying the above terms using the distributive law of multiplication, we get
\[ \Rightarrow {\left( {3m - 2n} \right)^3} = 27{m^3} - 8{n^3} - 18mn \times 3m - 18mn \times \left( { - 2n} \right)\]
Simplifying the expression, we get
\[ \Rightarrow {\left( {3m - 2n} \right)^3} = 27{m^3} - 8{n^3} - 54mn \times m + 36mn \times n\]
We know that if two terms with same bases and different exponents are multiplied, then the product is the same base with the power as the sum of the two exponents.
This can be written as \[{a^m} \times {a^n} = {a^{m \times n}}\].
Rewriting the equation \[{\left( {3m - 2n} \right)^3} = 27{m^3} - 8{n^3} - 54mn \times m + 36mn \times n\], we get
\[ \Rightarrow {\left( {3m - 2n} \right)^3} = 27{m^3} - 8{n^3} - 54{m^1}n \times {m^1} + 36m{n^1} \times {n^1}\]
Using the rule of exponents \[{a^m} \times {a^n} = {a^{m \times n}}\], we get
\[ \Rightarrow {\left( {3m - 2n} \right)^3} = 27{m^3} - 8{n^3} - 54{m^{1 + 1}}n + 36m{n^{1 + 1}}\]
Adding the terms in the exponents, we get
\[ \Rightarrow {\left( {3m - 2n} \right)^3} = 27{m^3} - 8{n^3} - 54{m^2}n + 36m{n^2}\]
Since there are no like terms in the above equation, we cannot simplify the expression further.
Thus, the value of the expression \[{\left( {3m - 2n} \right)^3}\] is \[27{m^3} - 8{n^3} - 54{m^2}n + 36m{n^2}\].

Note: We have used the distributive law of multiplication in the solution to multiply \[ - 18mn\] by \[3m - 2n\]. The distributive law of multiplication states that \[a\left( {b + c} \right) = a \cdot b + a \cdot c\].
We cannot simplify \[27{m^3} - 8{n^3} - 54{m^2}n + 36m{n^2}\] further because there are no like terms in the expression. Like terms are the terms whose variables as well as their exponents are the same. For example, \[100x,150x,240x,600x\] all have the variable \[x\] raised to the exponent 1. Terms which don’t have the same variables, or have different degrees of variables cannot be added together. For example, it is not possible to add \[27{m^3}\] to \[36m{n^2}\].