Final answer:
The expression 4x³((3)/(x⁻²)³) simplifies to 108x⁻³. We cube the number 3 to get 27 and multiply the exponent of x⁻⁶ by 3. Then we multiply 4 by 27 and combine the x terms by adding the exponents, which results in the simplified expression.
Step-by-step explanation:
To simplify the expression 4x³((3)/(x⁻²)³), we need to apply the rules of exponents.
Firstly, the parentheses indicated the power affects everything inside, so we cube the fraction (x⁻²)³. When cubing exponentials, we cube the digit and multiply the exponent by the power. So, (x⁻²)³ becomes x⁻²³, which simplifies further to x⁻⁶, considering exponent multiplication.
Secondly, the expression (3)³ is simply cubing the number 3, which results in 27.
Now, combine these pieces with 4x³: 4x³ * 27x⁻⁶
Multiplying the numeric coefficients 4 * 27 gives us 108.
Then, to combine the x terms, we add the exponents (3) + (-6) as per the rules of exponents for multiplication, which is x³ * x⁻⁶ = x⁻³.
Putting it all together, we get the final simplified expression 108x⁻³.