To solve the expression [tex]\(\sqrt[3]{-125 n^{12}}\)[/tex], let's break it down step by step.
### Step 1: Understanding the expression
The expression [tex]\(\sqrt[3]{-125 n^{12}}\)[/tex] represents the cube root of the product [tex]\(-125\)[/tex] and [tex]\(n^{12}\)[/tex].
### Step 2: Separating the terms
First, we write the expression as a product under the cube root:
[tex]\[
\sqrt[3]{-125 n^{12}} = \sqrt[3]{-125} \cdot \sqrt[3]{n^{12}}
\][/tex]
### Step 3: Calculating the cube root of [tex]\(-125\)[/tex]
Next, we find the cube root of [tex]\(-125\)[/tex]. The value is a complex number, since taking the cube root of a negative number isn't straightforward in real numbers.
The cube root of [tex]\(-125\)[/tex] is:
[tex]\[
\sqrt[3]{-125} = 2.5 + 4.330127018922192j
\][/tex]
### Step 4: Simplifying [tex]\(\sqrt[3]{n^{12}}\)[/tex]
Next, we simplify [tex]\(\sqrt[3]{n^{12}}\)[/tex]:
[tex]\[
\sqrt[3]{n^{12}} = \left( n^{12} \right)^{\frac{1}{3}}
\][/tex]
Using the property of exponents [tex]\((a^m)^n = a^{mn}\)[/tex], we simplify:
[tex]\[
\left( n^{12} \right)^{\frac{1}{3}} = n^{\frac{12}{3}} = n^4
\][/tex]
### Step 5: Combining the results
We now combine the two results obtained:
[tex]\[
\sqrt[3]{-125 n^{12}} = (2.5 + 4.330127018922192j) \cdot n^4
\][/tex]
Thus, the final answer is:
[tex]\[
\sqrt[3]{-125 n^{12}} = (2.5 + 4.330127018922192j) \cdot n^4
\][/tex]
In summary, the cube root of [tex]\(-125 n^{12}\)[/tex] is [tex]\((2.5 + 4.330127018922192j) \cdot n^4\)[/tex].
