Sure, let's solve the problem step-by-step:
To find the value of [tex]\(\sqrt[4]{81}\)[/tex] (the fourth root of 81), follow these steps:
1. Understand the Problem: You need to find a number [tex]\(x\)[/tex] such that when [tex]\(x\)[/tex] is raised to the power of 4, it equals 81. Mathematically, you are looking for [tex]\(x\)[/tex] such that [tex]\(x^4 = 81\)[/tex].
2. Prime Factorization: Begin with the number 81. You can simplify the process by breaking it down into its prime factors.
- 81 can be factorized as [tex]\( 81 = 3 \times 3 \times 3 \times 3 \)[/tex].
- Therefore, [tex]\(81 = 3^4\)[/tex].
3. Apply the Concept of Roots: Now, we recognize that we are looking for [tex]\(\sqrt[4]{81}\)[/tex]. Given that [tex]\(81 = 3^4\)[/tex], we can substitute:
- [tex]\(\sqrt[4]{81} = \sqrt[4]{3^4}\)[/tex].
4. Utilize Exponentiation Rules: When dealing with roots, the notation [tex]\(\sqrt[n]{a^m} = a^{m/n}\)[/tex] applies. So, in this case:
- [tex]\(\sqrt[4]{3^4} = 3^{4/4}\)[/tex].
5. Simplify the Exponent: Simplify the fraction in the exponent:
- [tex]\(3^{4/4} = 3^1\)[/tex].
6. Result: Simplifying [tex]\(3^1\)[/tex] gives us 3. Thus,
- [tex]\(\sqrt[4]{81} = 3\)[/tex].
Therefore, the fourth root of 81 is [tex]\( \boxed{3} \)[/tex].
