Write $(2x^3)^{3/4}$ as a radical expression

Key concept: Rational exponents follow the rule $a^{m/n}=\sqrt[n]{a^m}$. Here, the denominator $4$ tells you the root, and the numerator $3$ tells you the power.

Step by step

Use the rule

$a^{m/n}=\sqrt[n]{a^m}$

So

$(2x^3)^{3/4}=\sqrt[4]{(2x^3)^3}$

Now simplify inside the radical:

$(2x^3)^3=2^3x^9=8x^9$

Therefore,

$\boxed{(2x^3)^{3/4}=\sqrt[4]{8x^9}}$

If you want a form with a perfect fourth power pulled out, you can also write

$\sqrt[4]{8x^9}=x^2\sqrt[4]{8x}$

assuming $x\ge 0$ in a real-number context.

Pitfall alert

A common mistake is to put the exponent on only one part of the product, or to turn $\frac34$ into a cube root instead of a fourth root. Another frequent error is forgetting that $(x^3)^3=x^9$.

Try different conditions

If the expression were $(2x^3)^{1/4}$, then the radical form would be $\sqrt[4]{2x^3}$. If it were $(2x^3)^{5/4}$, then it would be $\sqrt[4]{(2x^3)^5}$.

Further reading

rational exponents, fourth root, radical expression