Certainly! Let's solve the given expression step-by-step:
### Given Expression:
[tex]\[ -\left(-2 x^3\right)^6 \][/tex]
### Step 1: Evaluate the Term Inside the Parentheses
First, consider the term inside the parentheses:
[tex]\[ (-2 x^3)^6 \][/tex]
### Step 2: Apply the Exponentiation
When raising a product to a power, we distribute the exponent to each factor in the product:
[tex]\[ (-2 x^3)^6 = (-2)^6 \cdot (x^3)^6 \][/tex]
#### Calculate the constants:
The term [tex]\((-2)^6\)[/tex] means raising [tex]\(-2\)[/tex] to the power of 6.
[tex]\[ (-2)^6 = 64 \][/tex]
#### Calculate the variable part:
The term [tex]\((x^3)^6\)[/tex] means raising [tex]\(x^3\)[/tex] to the power of 6.
[tex]\[ (x^3)^6 = x^{3 \cdot 6} = x^{18} \][/tex]
Combining these results, we have:
[tex]\[ 64 \cdot x^{18} \][/tex]
[tex]\[ (-2 x^3)^6 = 64 x^{18} \][/tex]
### Step 3: Apply the Negation
The given expression has a negation outside of the exponentiation:
[tex]\[ -\left( 64 x^{18} \right) \][/tex]
By distributing the negation, we get:
[tex]\[ -64 x^{18} \][/tex]
### Step 4: Compare with the Given Answer
The final term, as per the question, is given as:
[tex]\[ -12 x^9 \][/tex]
### Observation
The result from evaluating the expression of [tex]\(-(-2 x^3)^6\)[/tex] is:
[tex]\[ -64 x^{18} \][/tex]
However, based on the problem context, we're given that the final term should be:
[tex]\[ -12 x^9 \][/tex]
This indicates there might have been some mistake or possible additional context or condition that led to the given result. Without further context, purely from the algebraic manipulation, the expression simplifies to:
[tex]\[ -64 x^{18} \][/tex]
