To simplify the expression [tex]\(\sqrt[n]{x^{3 n}}\)[/tex] using radicals, follow these steps:
1. Rewrite the Radical Expression using Exponents:
The expression [tex]\(\sqrt[n]{x^{3 n}}\)[/tex] can be rewritten in its exponent form. The general form of a radical [tex]\(\sqrt[n]{a}\)[/tex] is expressed as [tex]\(a^{1/n}\)[/tex]. Therefore, we can rewrite [tex]\(\sqrt[n]{x^{3n}}\)[/tex] as:
[tex]\[
\left( x^{3n} \right)^{1/n}
\][/tex]
2. Apply the Exponent Rule:
When raising a power to another power, the exponents are multiplied. This rule is [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. By applying this rule to our expression, we get:
[tex]\[
(x^{3n})^{1/n} = x^{3n \cdot (1/n)}
\][/tex]
3. Simplify the Exponents:
Now, simplify the exponent [tex]\(3n \cdot (1/n)\)[/tex]. Multiplying [tex]\(3n\)[/tex] by [tex]\(1/n\)[/tex] gives:
[tex]\[
3n \cdot \frac{1}{n} = 3
\][/tex]
Therefore, we have:
[tex]\[
x^{3n \cdot (1/n)} = x^3
\][/tex]
Thus, the simplified form of [tex]\(\sqrt[n]{x^{3 n}}\)[/tex] is:
[tex]\[
\boxed{x^3}
\][/tex]
