To simplify the radical [tex]\(\sqrt{x^{13}}\)[/tex], we can follow these steps:
1. Express the radical as an exponent:
The square root of [tex]\(x^{13}\)[/tex] can be written as:
[tex]\[
\sqrt{x^{13}} = (x^{13})^{1/2}
\][/tex]
2. Use the property of exponents:
The property of exponents we will use is [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Applying this property, we get:
[tex]\[
(x^{13})^{1/2} = x^{13 \cdot \frac{1}{2}} = x^{13/2}
\][/tex]
3. Separate the exponent into a whole number part and a fractional part:
The exponent [tex]\(13/2\)[/tex] can be broken down as follows:
[tex]\[
x^{13/2} = x^{(6 + 1/2)} = x^6 \cdot x^{1/2}
\][/tex]
4. Express [tex]\(x^{1/2}\)[/tex] as a square root:
We know that [tex]\(x^{1/2} = \sqrt{x}\)[/tex]. Substituting this in:
[tex]\[
x^6 \cdot x^{1/2} = x^6 \cdot \sqrt{x}
\][/tex]
Thus, the simplified form of [tex]\(\sqrt{x^{13}}\)[/tex] is:
[tex]\[
x^6 \sqrt{x}
\][/tex]
So the answer is:
[tex]\[
x^6 \sqrt{x}
\][/tex]
