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\sqrt[9]{7^{3}}
Rewrite 343 as 7^{3}.
\left(7^{3}\right)^{\frac{1}{9}}
Convert from radical to exponential form.
7^{\frac{3}{9}}
To raise a power to another power, multiply the exponents.
7^{\frac{1}{3}}
Reduce the fraction \frac{3}{9} to lowest terms by extracting and canceling out 3.
\sqrt[3]{7}
Convert from exponential to radical form.