Certainly! Let's solve [tex]\(\sqrt{x^5}\)[/tex] step-by-step.
### Step 1: Understanding the Expression
We are given the expression [tex]\(\sqrt{x^5}\)[/tex]. This represents the square root of [tex]\(x^5\)[/tex].
### Step 2: Simplification Using Exponents
To simplify [tex]\(\sqrt{x^5}\)[/tex], we can re-write it in terms of exponents. Recall that the square root can be written as an exponent of [tex]\( \frac{1}{2} \)[/tex]. Therefore,
[tex]\[
\sqrt{x^5} = (x^5)^{\frac{1}{2}}
\][/tex]
### Step 3: Applying the Power Rule
Using the power rule for exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we can simplify the expression further. Here, we multiply the exponents [tex]\(5\)[/tex] and [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
(x^5)^{\frac{1}{2}} = x^{5 \cdot \frac{1}{2}}
\][/tex]
### Step 4: Simplifying the Exponents
Now, multiply the exponents [tex]\(5\)[/tex] and [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
5 \cdot \frac{1}{2} = \frac{5}{2}
\][/tex]
Therefore,
[tex]\[
(x^5)^{\frac{1}{2}} = x^{\frac{5}{2}}
\][/tex]
### Conclusion
After simplifying, we conclude that:
[tex]\[
\sqrt{x^5} = x^{\frac{5}{2}}
\][/tex]
So, the simplified form of [tex]\(\sqrt{x^5}\)[/tex] is [tex]\(x^{\frac{5}{2}}\)[/tex].
