To simplify the expression [tex]\(\left(343 x^2\right)^{\frac{2}{3}}\)[/tex], we'll need to follow several steps involving properties of exponents and roots. Let's break it down systematically:
1. Factor the base: Begin with the base inside the parentheses. Note that [tex]\(343\)[/tex] can be written as [tex]\(7^3\)[/tex]. So, we rewrite the term inside the parentheses as:
[tex]\[
343x^2 = 7^3 x^2
\][/tex]
2. Apply the exponent to the entire term: The expression then becomes:
[tex]\[
\left(7^3 x^2\right)^{\frac{2}{3}}
\][/tex]
3. Distribute the exponent: Using the property of exponents [tex]\((a \cdot b)^n = a^n \cdot b^n\)[/tex], we distribute [tex]\(\frac{2}{3}\)[/tex] to both [tex]\(7^3\)[/tex] and [tex]\(x^2\)[/tex]:
[tex]\[
\left(7^3\right)^{\frac{2}{3}} \cdot \left(x^2\right)^{\frac{2}{3}}
\][/tex]
4. Simplify each term separately:
- Simplify [tex]\(\left(7^3\right)^{\frac{2}{3}}\)[/tex]:
[tex]\[
\left(7^3\right)^{\frac{2}{3}} = 7^{3 \cdot \frac{2}{3}} = 7^2
\][/tex]
Since [tex]\(7^2 = 49\)[/tex], this part simplifies to [tex]\(49\)[/tex].
- Simplify [tex]\(\left(x^2\right)^{\frac{2}{3}}\)[/tex]:
[tex]\[
\left(x^2\right)^{\frac{2}{3}} = x^{2 \cdot \frac{2}{3}} = x^{\frac{4}{3}}
\][/tex]
5. Combine the simplified parts: Putting it all back together, we have:
[tex]\[
49 \cdot x^{\frac{4}{3}}
\][/tex]
Thus, the simplified form of [tex]\(\left(343 x^2\right)^{\frac{2}{3}}\)[/tex] is:
[tex]\[
49 x^{\frac{4}{3}}
\][/tex]
Therefore, the simplified expression is [tex]\(\boxed{49 x^{\frac{4}{3}}}\)[/tex].
