Rewrite the radical expression as an exponential expression: $\sqrt[3]{2x^8}$

Key concept: To convert a radical to an exponent, use the rule $\sqrt[n]{a^m}=a^{m/n}$. Keep the whole radicand under the same fractional exponent.

Step by step

Use the radical-to-exponent rule:

$\sqrt[n]{A}=A^{1/n}.$

So

$\sqrt[3]{2x^8}=(2x^8)^{1/3}.$

Distribute the exponent to each factor if desired:

$(2x^8)^{1/3}=2^{1/3}x^{8/3}.$

Either form is acceptable, but the direct exponential form is

$\boxed{(2x^8)^{1/3}}.$

Pitfall alert

Do not write $(2x)^{8/(-3)}$. The denominator should be positive when converting a cube root, and the radicand should stay inside the fractional exponent as a whole expression.

Try different conditions

If the radical were $\sqrt[5]{2x^8}$, the exponential form would be $(2x^8)^{1/5}$. If the radicand were just $x^8$, then the form would be $x^{8/3}$ after conversion.

Further reading

radical exponent rule, fractional power, cube root