Rewrite the exponential expression as a radical expression: $(3x+3)^{6/7}$

Key concept: Fractional exponents of the form $m/n$ convert to radicals by using the denominator as the root index and the numerator as the power.

Step by step

Use the rule

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

For

$(3x+3)^{6/7},$

the denominator $7$ means a seventh root, and the numerator $6$ stays as the power:

$(3x+3)^{6/7}=\sqrt[7]{(3x+3)^6}.$

So the radical form is

$\boxed{\sqrt[7]{(3x+3)^6}}.$

Pitfall alert

A common error is expanding $(3x+3)^6$ incorrectly or rewriting it as $\sqrt[7]{729x^6+729}$. The exponent $6$ applies to the entire binomial $3x+3$, not separately to each term inside the sum.

Try different conditions

If the expression were $(3x+3)^{1/7}$, the radical form would be $\sqrt[7]{3x+3}$. If it were $(3x+3)^{8/7}$, it would become $(3x+3)\sqrt[7]{3x+3}$ or $\sqrt[7]{(3x+3)^8}$ depending on the preferred form.

Further reading

fractional exponent, seventh root, power rule