Let's begin by finding a simplified form for the cube root of -128. To do this, we need to consider the properties of cube roots and the factors of -128.
### Step 1: Express -128 in terms of its prime factors.
We start by factoring 128:
[tex]\[ 128 = 2 \times 64 \][/tex]
[tex]\[ 64 = 2 \times 32 \][/tex]
[tex]\[ 32 = 2 \times 16 \][/tex]
[tex]\[ 16 = 2 \times 8 \][/tex]
[tex]\[ 8 = 2 \times 4 \][/tex]
[tex]\[ 4 = 2 \times 2 \][/tex]
So, we have:
[tex]\[ 128 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \][/tex]
[tex]\[ 128 = 2^7 \][/tex]
### Step 2: Incorporate the negative sign.
Since we have -128, we can write:
[tex]\[ -128 = -2^7 \][/tex]
### Step 3: Apply the cube root.
We need to take the cube root of -2^7:
[tex]\[ \sqrt[3]{-2^7} = \sqrt[3]{- (2^7)} \][/tex]
### Step 4: Simplify the cube root.
Recall that:
[tex]\[ \sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b} \][/tex]
Thus:
[tex]\[ \sqrt[3]{-2^7} = \sqrt[3]{-1 \cdot 2^7} = \sqrt[3]{-1} \cdot \sqrt[3]{2^7} \][/tex]
### Step 5: Evaluate the cube root of -1.
The cube root of -1 is -1:
[tex]\[ \sqrt[3]{-1} = -1 \][/tex]
### Step 6: Evaluate the cube root of [tex]\(2^7\)[/tex].
We can split 2^7 in a form that's easy to handle with cube roots:
[tex]\[ 2^7 = 2^{3+3+1} = (2^3) \cdot (2^3) \cdot 2 \][/tex]
[tex]\[ \sqrt[3]{2^7} = \sqrt[3]{(2^3) \cdot (2^3) \cdot 2} = 2 \cdot 2 \cdot \sqrt[3]{2} = 4 \sqrt[3]{2} \][/tex]
### Step 7: Combine the results.
Combining the results from steps 5 and 6:
[tex]\[ \sqrt[3]{-128} = -1 \cdot 4 \sqrt[3]{2} = -4 \sqrt[3]{2} \][/tex]
### Conclusion:
The expression simplified is:
[tex]\[ \sqrt[3]{-128} = -4 \sqrt[3]{2} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{-4 \sqrt[3]{2}} \][/tex]
The corresponding option in the given choices:
A. [tex]\( -4 \sqrt[3]{2} \)[/tex]
