To determine the value of the expression [tex]\(-4 \sqrt[3]{-6 m^2}\)[/tex] when [tex]\( m = 6 \)[/tex], we begin by substituting the value of [tex]\( m \)[/tex] into the expression.
Given [tex]\( m = 6 \)[/tex], we first find:
[tex]\[ -6 m^2 = -6 (6^2) = -6 (36) = -216 \][/tex]
Next, we need to calculate the cube root of [tex]\(-216\)[/tex]:
[tex]\[ \sqrt[3]{-216} \][/tex]
The cube root of [tex]\(-216\)[/tex] is a complex number because the cube root of a negative number involves imaginary parts. Specifically:
[tex]\[ \sqrt[3]{-216} = -6 (1 + 0j) \][/tex]
Here, [tex]\(-6\)[/tex] is one of the principal cube roots of [tex]\(-216\)[/tex]. We now multiply this result by [tex]\(-4\)[/tex]:
[tex]\[ -4 \times \sqrt[3]{-216} = -4 \times (-6) \][/tex]
This includes multiplying by the cyclic roots of unity. The complete result is:
[tex]\[ -4 \times \big( -6(1 + 0.86602540378i)\big) = -4 \big(-6 -12.4722463 i\big)\][/tex]
Thus, the final value of the expression is:
[tex]\[ (-12 - 20.784609690826525j) \][/tex]
The numerical complex number [tex]\((-12 - 20.784609690826525 j)\)[/tex] is the exact evaluation for the given expression.
Therefore, the correct answer does not match any of the given options as the solution involves a complex number.
