To simplify the expression [tex]\(\sqrt{x^{17}}\)[/tex], we will use the properties of exponents and radicals. Let's go through the steps in detail:
1. Rewrite the Radical Expression:
First, recognize that the square root of a number can be expressed as that number raised to the power of [tex]\( \frac{1}{2} \)[/tex]. Therefore, we can rewrite [tex]\(\sqrt{x^{17}}\)[/tex] as:
[tex]\[
\sqrt{x^{17}} = (x^{17})^{\frac{1}{2}}
\][/tex]
2. Apply the Power of a Power Rule:
The rule for exponents that states [tex]\((a^m)^n = a^{m \cdot n}\)[/tex] applies here. We will apply this rule to simplify the expression:
[tex]\[
(x^{17})^{\frac{1}{2}} = x^{17 \cdot \frac{1}{2}}
\][/tex]
3. Perform the Multiplication in the Exponent:
Multiply the exponents [tex]\(17\)[/tex] and [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
17 \cdot \frac{1}{2} = \frac{17}{2}
\][/tex]
4. Rewrite the Expression:
Now, rewrite the expression with the simplified exponent:
[tex]\[
x^{\frac{17}{2}}
\][/tex]
Therefore, the simplified form of [tex]\(\sqrt{x^{17}}\)[/tex] is:
[tex]\[
\boxed{x^{\frac{17}{2}}}
\][/tex]
