Root of a Root - Examples, Exercises and Solutions

To simplify the given expression, we will use two laws of exponents:

A. Definition of the root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

B. Law of exponents for an exponent on an exponent:

$(a^m)^n=a^{m\cdot n}$

Let's begin simplifying the given expression:

$\sqrt[4]{\sqrt[3]{3}}= \\$We will use the law of exponents shown in A and first convert the roots in the expression to exponents, we will do this in two steps - in the first step we will convert the inner root in the expression and in the next step we will convert the outer root:

$\sqrt[4]{\sqrt[3]{3}}= \\ \sqrt[4]{3^{\frac{1}{3}}}= \\ (3^{\frac{1}{3}})^{\frac{1}{4}}=$We continue and use the law of exponents shown in B, then we will multiply the exponents:

$(3^{\frac{1}{3}})^{\frac{1}{4}}= \\ 3^{\frac{1}{3}\cdot\frac{1}{4}}=\\ 3^{\frac{1\cdot1}{3\cdot4}}=\\ \boxed{3^{\frac{1}{12}}}=\\ \boxed{\sqrt[12]{3}}$ In the final step we return to writing the root, that is - back, using the law of exponents shown in A (in the opposite direction),

Let's summarize the simplification of the given expression:

$\sqrt[4]{\sqrt[3]{3}}= \\ (3^{\frac{1}{3}})^{\frac{1}{4}}= \\ \boxed{3^{\frac{1}{12}}}=\\ \boxed{\sqrt[12]{3}}$Therefore, note that the correct answer (most) is answer D.