Square Roots

Let's use the definition of root as a power:

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

when we remember that in a square root (also called "root to the power of 2") we don't write the root's power and:

$n=2$

meaning:

$\sqrt{a}=\sqrt[2]{a}=a^{\frac{1}{2}}$

Let's return to the problem and convert using the root definition we mentioned above the roots in the problem:

$\sqrt[6]{\sqrt{2}}=\sqrt[6]{2^{\frac{1}{2}}}=\big(2^{\frac{1}{2}}\big)^{\frac{1}{6}}$

where in the first stage we applied the root definition as a power mentioned earlier to the inner expression (meaning inside the larger-outer root) and then we used parentheses and applied the same definition to the outer root.

Now let's remember the power law for power of a power:

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

Let's apply this law to the expression we got in the last stage:

$\big(2^{\frac{1}{2}}\big)^{\frac{1}{6}}=2^{\frac{1}{2}\cdot\frac{1}{6}}=2^{\frac{1\cdot1}{2\cdot6}}=2^{\frac{1}{12}}$

where in the first stage we applied the power law mentioned above and then simplified the resulting expression and performed the multiplication of fractions in the power exponent.

Let's summarize the solution steps so far, we got that:

$\sqrt[6]{\sqrt{2}}=\big(2^{\frac{1}{2}}\big)^{\frac{1}{6}} =2^{\frac{1}{12}}$

In the next stage we'll apply again the root definition as a power that was mentioned at the beginning of the solution, but in the opposite direction:

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

Let's apply this law to go back and present the expression we got in the last stage in root form:

$$$2^{\frac{1}{12}} =\sqrt[12]{2}$

Therefore we got that:

$\sqrt[6]{\sqrt{2}}=2^{\frac{1}{12}} =\sqrt[12]{2}$

Therefore the correct answer is answer A.

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.