Let's analyze and solve the given equation step-by-step.
We start with the expression:
[tex]\[ -\left( -2x^3 \right)^6 \][/tex]
Step 1: Expand the inner power expression.
First, consider [tex]\(\left( -2x^3 \right)^6\)[/tex]. When raising a product to a power, each term inside the parentheses is raised to that power:
[tex]\[ \left( -2x^3 \right)^6 = (-2)^6 \cdot (x^3)^6 \][/tex]
Step 2: Simplify the constants and the variable powers.
First, calculate [tex]\((-2)^6\)[/tex]:
[tex]\[ (-2)^6 = (-2) \cdot (-2) \cdot (-2) \cdot (-2) \cdot (-2) \cdot (-2) = 64 \][/tex]
Next, calculate [tex]\((x^3)^6\)[/tex]:
[tex]\[ (x^3)^6 = x^{3 \cdot 6} = x^{18} \][/tex]
So, combining these results, we have:
[tex]\[ \left( -2x^3 \right)^6 = 64x^{18} \][/tex]
Step 3: Apply the negative sign outside the parentheses.
Since the expression started with a negative sign outside the parentheses, we have:
[tex]\[ -\left( 64x^{18} \right) = -64x^{18} \][/tex]
Thus,
[tex]\[ -\left( -2x^3 \right)^6 = -64x^{18} \][/tex]
Now, considering the final simplified part of the problem [tex]\( -12x^9 \)[/tex]:
We already have:
[tex]\[ -64x^{18} \][/tex]
To interpret this:
[tex]\[ -64x^{18} \][/tex]
is the expanded form of the expression.
So the equation given to satisfy is:
[tex]\[ -64x^{18} = -12x^9 \][/tex]
The solution in this situation indicates that we have reworked and simplified an expression starting from [tex]\( -\left( -2x^3 \right)^6 \)[/tex] showing its expanded form as [tex]\( 64x^{18} \)[/tex]. Both must be correct representations and what they ask must meet or simplify to [tex]\(-12x^9\)[/tex].
Thus our final components are:
[tex]\[ -64x^{18} \quad \text{represents the left side expanded form}\][/tex]
[tex]\[ -12x^9 \quad \text{is the simplified given final part noted} \][/tex]
