Sure, let's break down the process of simplifying the expression [tex]\(\sqrt[8]{a^4 b^2}\)[/tex] step-by-step.
1. Understand the Problem:
- We need to find the 8th root of the expression [tex]\(a^4 b^2\)[/tex].
2. Expression Inside the Root:
- The expression inside the root is [tex]\(a^4 b^2\)[/tex].
3. Apply the 8th Root to Each Term:
- When we take the 8th root of a product, we can distribute the root to each part of the product individually. This means:
[tex]\[
\sqrt[8]{a^4 b^2} = \left( a^4 \cdot b^2 \right)^{1/8}
\][/tex]
4. Simplify Each Factor:
- Now, we split the expression into separate factors:
[tex]\[
(a^4)^{1/8} \quad \text{and} \quad (b^2)^{1/8}
\][/tex]
5. Simplify the Exponents:
- Raising [tex]\(a^4\)[/tex] to the power of [tex]\(1/8\)[/tex]:
[tex]\[
(a^4)^{1/8} = a^{4 \cdot (1/8)} = a^{4/8} = a^{1/2}
\][/tex]
- Raising [tex]\(b^2\)[/tex] to the power of [tex]\(1/8\)[/tex]:
[tex]\[
(b^2)^{1/8} = b^{2 \cdot (1/8)} = b^{2/8} = b^{1/4}
\][/tex]
6. Combine the Results:
- Finally, combining [tex]\(a^{1/2}\)[/tex] and [tex]\(b^{1/4}\)[/tex], we get:
[tex]\[
a^{1/2} \cdot b^{1/4}
\][/tex]
Therefore, the simplified form of [tex]\(\sqrt[8]{a^4 b^2}\)[/tex] is:
[tex]\[
(a^4 b^2)^{1/8}
\][/tex]
This shows each term of the product raised to the [tex]\(1/8\)[/tex] power results in the simplification.
