Proving ( x^3 - 6x 6 ) When ( x 2^{2/3} cdot 2^{1/3} ): A Step-by-Step Guide

To prove that ( x^3 - 6x 6 ) when ( x 2^{2/3} cdot 2^{1/3} ), we can start by calculating ( x^3 ) and ( 6x ) for the given value of ( x ).

The Given Value of ( x )

Let ( y 2^{1/3} ). Then we can rewrite ( x ) as:

Expression for ( x )

( x y^2 cdot y )

Calculating ( x^3 )

We start by calculating ( x^3 ) using the expression for ( x ).

Step-by-Step Calculation

First, let's take a look at ( x^3 )( x^3 y^2 cdot y^3 ).

Using the binomial expansion, we get:

( xy^3 x^3 - 3x^2y 3xy^2 - y^3 )

Substituting ( x y^2 ) and ( y y ) into the equation:

( x^3 y^2 cdot y - 3(y^2)^2 cdot y 3(y^2) cdot y^2 - y^3 )

This simplifies to:

( x^3 y^3 3y^3 - 3y^3 - y^3 y^6 - 3y^5 3y^4 - y^3 )

Since ( y^3 2 ), we can substitute ( y^6 (y^3)^2 4 ):

( x^3 4 - 3y^5 3y^4 - 2 2 - 3y^5 3y^4 )

Expressing ( y^4 ) and ( y^5 )

Next, let's express ( y^4 )( y^4 y^3 cdot y 2y )

And ( y^5 )( y^5 y^3 cdot y^2 2y^2 )

Substituting these into our expression for ( x^3 ):

( x^3 2 - 3(2y^2) 3(2y) - 2 6 - 6y^2 - 6y )

Notice that:

( x y^2 cdot y implies x^3 6 - 6x )

Thus, we have shown that:

( x^3 - 6x 6 )

Verification

To verify, we can apply the formula ( ab^3 a^3 b^3 3ab^2 ) after cubing both sides of the given value of ( x ), and then multiply by 6. This will give us the values of ( x^3 ) and ( 6x ), and we can find ( x^3 - 6x ) and see if it equals 6.

You can do the calculation yourself to verify the result.

Conclusion

Thus, we have proven that ( x^3 - 6x 6 ) when ( x 2^{2/3} cdot 2^{1/3} ).