Square Roots

To solve the expression $\sqrt{\sqrt{4}}$, we'll proceed with the following steps:

• Step 1: Evaluate the inner square root.
  The expression $\sqrt{4}$ simplifies to 2, because 2 squared is 4.
• Step 2: Now evaluate the square root of 2.
  Since the result from step 1 is 2, we need to find $\sqrt{2}$. This is the prime representation of the result because 2 cannot be further simplified.

Therefore, the answer to the problem $\sqrt{\sqrt{4}}$ is $\sqrt{2}$.

In order to solve the following expression $\sqrt{\sqrt{12}}$, it needs to be simplified using the properties of exponents and roots. Specifically, we apply the rule that states that the square root of a square root can be expressed as a fourth root.

Let's break down this solution step by step:

• First, represent the inner $\sqrt{12}$ as a power: $12^{1/2}$.

• Next, take the square root of this result, which involves raising $12^{1/2}$ to the power of $1/2$ again:
  $\left(12^{1/2}\right)^{1/2} = 12^{(1/2) \cdot (1/2)} = 12^{1/4}$.

• According to the rules of exponents, raising an exponent to another power results in multiplying the exponents.

• This gives us $12^{1/4}$, which we can write as the fourth root of 12: $\sqrt[4]{12}$.

In conclusion the simplification of $\sqrt{\sqrt{12}}$ is $\sqrt[4]{12}$.

In order to solve this problem, we must simplify the following expression $\sqrt[6]{\sqrt{2}}$ using the rule for roots of roots. This rule states that a root of a root can be written as a single root by multiplying the indices of the radicals.

• Step 1: Identify the given expression $\sqrt[6]{\sqrt{2}}$.

• Step 2: Recognize that the inner root, $\sqrt{2}$, can be expressed as $\sqrt[2]{2}$.

• Step 3: Visualize $\sqrt[6]{\sqrt{2}}$ as $\sqrt[6]{\sqrt[2]{2}}$.

• Step 4: Apply the rule $\sqrt[n]{\sqrt[m]{a}} = \sqrt[n \times m]{a}$.

• Step 5: Multiply the indices: $6 \times 2 = 12$.

• Step 6: Replace the compound root with the single root: $\sqrt[12]{2}$.

Thus, the expression $\sqrt[6]{\sqrt{2}}$ simplifies to $\sqrt[12]{2}$.

Therefore, the solution to the problem is $\sqrt[12]{2}$.

In order to solve the given problem, we'll follow these steps:

• Step 1: Convert the inner square root to an exponent: $\sqrt{8} = 8^{1/2}$.

• Step 2: Apply the root of a root property: $\sqrt{\sqrt{8}} = (\sqrt{8})^{1/2} = (8^{1/2})^{1/2}$.

• Step 3: Simplify the expression using exponent rules: $(8^{1/2})^{1/2} = 8^{(1/2) \cdot (1/2)} = 8^{1/4}$.

The nested root expression simplifies to $8^{1/4}$.

Therefore, the simplified expression of $\sqrt{\sqrt{8}}$ is $8^{\frac{1}{4}}$.

After comparing this result with the multiple choice answers, choice 2 is correct.

Express the definition of root as a power:

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

Remember that in a square root (also called "root to the power of 2") we don't write the root's power as shown below:

$n=2$

Meaning:

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

Now convert the roots in the problem using the root definition provided above. :

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

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.

Let's recall the power law for power of a power:

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

Apply this law to the expression that we obtained in the last stage:

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

In the first stage we applied the power law mentioned above and then proceeded first to simplify the resulting expression and then to perform the multiplication of fractions in the power exponent.

Let's summarize the various steps of the solution thus far:

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

In the next stage we'll apply once again the root definition as a power, (that was mentioned at the beginning of the solution) however this time in the opposite direction:

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

Let's apply this law in order to present the expression we obtained in the last stage in root form:

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

We obtain the following result: :

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

Therefore the correct answer is answer A.