To simplify the expression [tex]\(\sqrt{x^8}\)[/tex] where [tex]\(x\)[/tex] represents a positive real number, we can follow these steps:
1. Understand the Problem: We need to simplify the expression [tex]\(\sqrt{x^8}\)[/tex].
2. Use the Property of Square Roots: Recall that the square root of a variable raised to a power can be expressed as the variable raised to half of that power. Specifically, [tex]\(\sqrt{a^b} = a^{b/2}\)[/tex].
3. Apply the Property:
- Here, our expression is [tex]\(\sqrt{x^8}\)[/tex].
- Using the property from step 2, we rewrite [tex]\(\sqrt{x^8}\)[/tex] as [tex]\(x^{8/2}\)[/tex].
4. Simplify the Exponent:
- [tex]\(8/2 = 4\)[/tex].
- Therefore, [tex]\(x^{8/2} = x^4\)[/tex].
So, the simplified form of [tex]\(\sqrt{x^8}\)[/tex] is:
[tex]\[
\sqrt{x^8} = x^4
\][/tex]
Thus, the simplified expression is [tex]\(x^4\)[/tex].
