For an expression like $\sqrt[n]{m^k}$, try rewriting it as $m^{k/n}$. Do this for each answer choice so you can compare all of them in the same format.

Once each option is written as $m^{\text{(fraction)}}$, which one has the same exponent as $m^{5/6}$?

In Desmos, type f(x) = x^(5/6) (this represents $x^{5/6}$ for $x>0$).

Create separate functions for each choice, for example:

• gA(x) = nthroot(x^25, 10) for $\sqrt[10]{x^{25}}$
• gB(x) = nthroot(x^5, 6) for $\sqrt[6]{x^{5}}$
• gC(x) = nthroot(x^35, 12) for $\sqrt[12]{x^{35}}$
• gD(x) = nthroot(x^10, 3) for $\sqrt[3]{x^{10}}$

Look at the graphs of $f(x)$ and each $g(x)$ for $x>0$. The function whose graph lies exactly on top of $f(x) = x^(5/6)$ for all shown $x>0$ corresponds to the equivalent expression in the answer choices.

A fractional exponent means a root. In general, for any positive number $a$ and integers $p$ and $q>0$:

• $a^{p/q} = \sqrt[q]{a^p}$.

So to compare expressions, it is helpful to rewrite radicals as powers (exponents) or powers as radicals using this rule.

Use $\sqrt[n]{m^k} = m^{k/n}$ to turn each choice into a power of $m$.

• Choice A: $\sqrt[10]{m^{25}} = m^{25/10} = m^{5/2}$.
• Choice B: $\sqrt[6]{m^{5}} = m^{5/6}$.
• Choice C: $\sqrt[12]{m^{35}} = m^{35/12}$.
• Choice D: $\sqrt[3]{m^{10}} = m^{10/3}$.

Now every option is written as $m^{\text{(some exponent)}}$.

The original expression is $m^{5/6}$, so we need the choice whose exponent is exactly $5/6$.

From the exponents:

• A has exponent $5/2$.
• B has exponent $5/6$.
• C has exponent $35/12$.
• D has exponent $10/3$.

Only choice B has exponent $5/6$, so the equivalent expression is choice B, $\sqrt[6]{m^{5}}$.