Sure! Let's solve the expression [tex]\(\sqrt{x^5}\)[/tex] step by step.
1. Understanding the Expression:
The expression [tex]\(\sqrt{x^5}\)[/tex] involves a square root and a power.
2. Expressing Using Fractional Exponents:
Recall that the square root of a number can be written using a fractional exponent. Specifically, [tex]\(\sqrt{a} = a^{1/2}\)[/tex].
Therefore, [tex]\(\sqrt{x^5}\)[/tex] can be written as:
[tex]\[
\sqrt{x^5} = (x^5)^{1/2}
\][/tex]
3. Applying the Power of a Power Rule:
When you take a power of a power in exponents, you multiply the exponents. So, using the rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we apply this to our expression:
[tex]\[
(x^5)^{1/2} = x^{5 \cdot \frac{1}{2}}
\][/tex]
4. Simplification:
Now, multiply the exponents:
[tex]\[
5 \cdot \frac{1}{2} = \frac{5}{2}
\][/tex]
Thus,
[tex]\[
x^{5 \cdot \frac{1}{2}} = x^{\frac{5}{2}}
\][/tex]
5. Rewriting the Answer:
We can leave the final answer in the exponent form or convert it back to a root form if desired. In this case, the simplified form is:
[tex]\[
x^{\frac{5}{2}}
\][/tex]
Therefore, the expression [tex]\(\sqrt{x^5}\)[/tex] simplifies to:
[tex]\[
\sqrt{x^5} = x^{\frac{5}{2}}
\][/tex]
However, as stated, the form [tex]\(\sqrt{x^5}\)[/tex] itself is a common way to represent solutions in radical notation.
So, in summary:
[tex]\[
\sqrt{x^5} = \sqrt{x^5}
\][/tex]
or in exponent form:
[tex]\[
\sqrt{x^5} = x^{\frac{5}{2}}
\][/tex]
Both forms are widely accepted depending upon the context in which you need to use them.
