To solve the expression [tex]\(\sqrt[5]{-32 a^5}\)[/tex], let's break it down step-by-step:
1. Understanding the Expression:
The expression [tex]\(\sqrt[5]{-32 a^5}\)[/tex] asks us to find the fifth root of [tex]\(-32 a^5\)[/tex].
2. Breaking Down the Components:
- [tex]\(-32\)[/tex] is a constant term.
- [tex]\(a^5\)[/tex] is the variable part raised to the power of 5.
3. Finding the Fifth Root of the Constant:
Let's find the fifth root of [tex]\(-32\)[/tex]:
- We know that [tex]\(32\)[/tex] is [tex]\(2^5\)[/tex].
- Hence, [tex]\(-32\)[/tex] can be written as [tex]\((-2)^5\)[/tex].
- The fifth root of [tex]\((-2)^5\)[/tex] is [tex]\(-2\)[/tex].
4. Finding the Fifth Root of the Variable:
- We have [tex]\(a^5\)[/tex].
- The fifth root of [tex]\(a^5\)[/tex] is [tex]\(a\)[/tex].
5. Combining the Results:
Now, we combine the fifth roots of the constant and the variable:
- The fifth root of [tex]\(-32 a^5\)[/tex] can thus be written as [tex]\((-2) a\)[/tex].
Therefore, the fifth root of [tex]\(-32 a^5\)[/tex] is [tex]\(2.0 \cdot (-a^5)^{0.2}\)[/tex].
So, the expression simplifies to:
[tex]\[ \sqrt[5]{-32 a^5} = 2.0 \cdot (-a^5)^{0.2} \][/tex]
