How do I simplify the expression \[{\left( {{x^2} - 1} \right)^3}\]?

Hint: In the given question, we have been given an algebraic expression. It is clearly a cube. So we simply have to simplify a cubic expression. To do that, we just apply the square formula over the expression and we solve it, and we will have our answer.

Formula Used:
We are going to use the cube formula:
\[{\left( {a - b} \right)^3} = {a^3} - {b^3} - 3ab\left( {a - b} \right) = {a^3} - {b^3} - 3{a^2}b + 3a{b^2}\]

Complete step-by-step answer:
We have to simplify the value of \[{\left( {{x^2} - 1} \right)^3}\].
We are going to use the square formula:
\[{\left( {a - b} \right)^3} = {a^3} - {b^3} - 3ab\left( {a - b} \right) = {a^3} - {b^3} - 3{a^2}b + 3a{b^2}\]
Substituting the value of \[a = {x^2}\] and \[b = 1\], we get,
\[{\left( {{x^2} - 1} \right)^3} = {\left( {{x^2}} \right)^3} - {1^3} - 3 \times {\left( {{x^2}} \right)^2} \times 1 + 3 \times \left( {{x^2}} \right) \times 1\]
Simplifying,
\[ = {x^6} - 3{x^4} + 3{x^2} - 1\]
Hence, \[{\left( {{x^2} - 1} \right)^3} = {x^6} - 3{x^4} + 3{x^2} - 1\]

Additional Information:
Here, we used the difference of two numbers whole cube formula, but if there was a plus sign, then we would have used the sum of two numbers whole cube formula, which is:
\[{\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right) = {a^3} + {b^3} + 3{a^2}b + 3a{b^2}\]

Note: In the given question, we simply had to put the formula of difference of two numbers in the whole squared formula. Then we just substituted the values, simplified the result and we got our answer. So, it is really important that we know the formulae and where, when and how to use them so that we can get the correct result.