Root of a Root: Identify the greater value

We need to find the largest of the given roots of 64:

• Calculate $\sqrt[6]{64}$:
  $\sqrt[6]{64} = 64^{1/6}$
  Since $64 = 2^6$, we have:
  $(2^6)^{1/6} = 2$

• Calculate $\sqrt[4]{64}$:
  $\sqrt[4]{64} = 64^{1/4}$
  Using the exponent $64 = 2^6$, we get:
  $(2^6)^{1/4} = 2^{6/4} = 2^{1.5} = \sqrt[2]{8} = 2 \times \sqrt{2} \approx 2.828$

• Calculate $\sqrt[3]{64}$:
  $\sqrt[3]{64} = 64^{1/3}$
  This simplifies to:
  $(2^6)^{1/3} = 2^{6/3} = 2^2 = 4$

• Calculate $\sqrt[]{64}$:
  $\sqrt[]{64} = 64^{1/2}$
  This gives us:
  $(2^6)^{1/2} = 2^{6/2} = 2^3 = 8$

Now, let's compare these calculated values:
- $\sqrt[6]{64} = 2$
- $\sqrt[4]{64} \approx 2.828$
- $\sqrt[3]{64} = 4$
- $\sqrt[]{64} = 8$

Among these values, the largest value is $\sqrt[]{64}$, which equals 8.

Therefore, the largest value is $\sqrt[]{64}$.

To solve this problem, we'll simplify each expression involving the roots of 1:

• Simplify $\sqrt[10]{\sqrt[10]{1}}$:
  Since $\sqrt[10]{1} = 1$, then $\sqrt[10]{\sqrt[10]{1}} = \sqrt[10]{1} = 1$.
• Simplify $\sqrt[10]{\sqrt[3]{1}}$:
  Since $\sqrt[3]{1} = 1$, then $\sqrt[10]{\sqrt[3]{1}} = \sqrt[10]{1} = 1$.
• Simplify $\sqrt[5]{\sqrt{1}}$:
  Since $\sqrt{1} = 1$, then $\sqrt[5]{\sqrt{1}} = \sqrt[5]{1} = 1$.

Upon simplifying, each of the options results in the value 1. Therefore, all expressions are equal.

The correct answer is: "All answers are correct".

To solve this problem, we'll follow these steps:

• Step 1: Convert each expression into exponential form.
• Step 2: Compare the resulting expressions by examining the exponents.
• Step 3: Identify which expression corresponds to the largest value.

Let's work through the solution:

Step 1: Convert each root to exponential form:
- $\sqrt{2} = 2^{0.5}$
- $\sqrt[3]{2} = 2^{1/3} \approx 2^{0.333}$
- $\sqrt[4]{2} = 2^{1/4} \approx 2^{0.25}$
- $\sqrt[5]{2} = 2^{1/5} \approx 2^{0.2}$

Step 2: Compare the exponents $0.5$, $0.333$, $0.25$, and $0.2$. Clearly, $0.5$ is the largest among these values.

Step 3: The expression with the largest exponent is $\sqrt{2} = 2^{0.5}$, so $\sqrt{2}$ is the largest value.

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

To determine if one of these values is the largest or if they are equal, we will express each expression as a power of 5:

• First, consider $\sqrt[6]{\sqrt{5}}$. This is equivalent to $(\sqrt{5})^{1/6} = (5^{1/2})^{1/6}$. Applying the power rule, this is $5^{1/12}$.
• Next, consider $\sqrt[12]{5}$, which is already expressed as a power: $5^{1/12}$.
• Finally, consider $\sqrt[4]{\sqrt[3]{5}}$. This can be rewritten as $(\sqrt[3]{5})^{1/4} = (5^{1/3})^{1/4}$. Again, using the power rule, this is $5^{1/12}$.

All three expressions simplify to $5^{1/12}$. Therefore, all values are equal. The correct choice is:

All values are equal.

To solve this problem, we'll follow the steps below:

• Simplify each mathematical expression using exponent rules.
• Compare the values derived from each simplification.

Let us analyze each given choice:

Choice 1: $\sqrt{\sqrt[6]{4}}$

• The expression is simplified as follows: $(\sqrt[6]{4})^{\frac{1}{2}} = 4^{\frac{1}{6} \times \frac{1}{2}} = 4^{\frac{1}{12}}$.

Choice 2: $\sqrt[6]{4}$

• This expression is: $4^{\frac{1}{6}}$.

Choice 3: $\sqrt[2]{\sqrt[3]{4}}$

• Simplified, this is: $(\sqrt[3]{4})^{\frac{1}{2}} = 4^{\frac{1}{3} \times \frac{1}{2}} = 4^{\frac{1}{6}}$.

Choice 4: $\sqrt{4}$

• This expression is equivalent to: $4^{\frac{1}{2}}$.

Now, let's compare the powers of 4:

• Choice 1: $4^{\frac{1}{12}}$
• Choice 2: $4^{\frac{1}{6}}$
• Choice 3: $4^{\frac{1}{6}}$
• Choice 4: $4^{\frac{1}{2}}$ - The largest exponent

The largest value among the given choices occurs when the exponent applied to the base 4 is maximized. Thus, the largest value is $\sqrt{4}$.

To solve this problem, we need to express each nested root as a power of 2:

• For $\sqrt{\sqrt{2}}$:
  $\sqrt{\sqrt{2}} = 2^{1/4}$
• For $\sqrt[3]{\sqrt{2}}$:
  $\sqrt[3]{\sqrt{2}} = 2^{1/6}$
• For $\sqrt[4]{\sqrt{2}}$:
  $\sqrt[4]{\sqrt{2}} = 2^{1/8}$
• For $\sqrt[5]{\sqrt{2}}$:
  $\sqrt[5]{\sqrt{2}} = 2^{1/10}$

Now, we compare these powers:

• $2^{1/4} > 2^{1/6} > 2^{1/8} > 2^{1/10}$.

Therefore, the largest value is $\sqrt{\sqrt{2}}$, which corresponds to choice 1.

To solve this problem, we'll proceed by evaluating each expression separately:

• Step 1: Evaluate $\sqrt{36}$.
• Step 2: Evaluate $\sqrt[3]{216}$.
• Step 3: Evaluate $\sqrt{\sqrt{1296}}$.
• Step 4: Compare the results to find the largest value.

Now, let's work through each step:
Step 1: Evaluate $\sqrt{36}$.
Since $6^2 = 36$, it follows that $\sqrt{36} = 6$.

Step 2: Evaluate $\sqrt[3]{216}$.
Since $6^3 = 216$, it follows that $\sqrt[3]{216} = 6$.

Step 3: Evaluate $\sqrt{\sqrt{1296}}$.
First, find $\sqrt{1296}$. Since $36^2 = 1296$, $\sqrt{1296} = 36$.
Then, find $\sqrt{36}$. Using Step 1, $\sqrt{36} = 6$.
Thus, $\sqrt{\sqrt{1296}} = 6$.

Step 4: Compare the results. We find that:
$\sqrt{36} = 6$
$\sqrt[3]{216} = 6$
$\sqrt{\sqrt{1296}} = 6$

Since all three values are equal, each expression evaluates to 6. The answer choice that states "All answers are correct" is indeed correct. Therefore, the solution to the problem is "All answers are correct."