Root of a Root: Applying the formula

To solve the problem of finding $\sqrt[5]{\sqrt[3]{5}}$, we'll use the formula for a root of a root, which combines the exponents:

• Step 1: Express each root as an exponent.
  We start with the innermost root: $\sqrt[3]{5} = 5^{1/3}$.
• Step 2: Apply the outer root.
  The square root to the fifth power is expressed as: $\sqrt[5]{5^{1/3}} = (5^{1/3})^{1/5}$.
• Step 3: Combine the exponents.
  Using the exponent rule $(a^m)^n = a^{m \times n}$, we get:
  $(5^{1/3})^{1/5} = 5^{(1/3) \times (1/5)} = 5^{1/15}$.
• Step 4: Convert the exponent back to root form.
  This can be written as $\sqrt[15]{5}$.

Therefore, the simplified expression of $\sqrt[5]{\sqrt[3]{5}}$ is $\sqrt[15]{5}$.

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 $\sqrt{2}$.

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 solve this problem, we'll simplify the 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 $\sqrt[12]{2}$.

To solve the expression $\sqrt{\sqrt{12}}$, we simplify 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: $\sqrt[4]{12}$.

We conclude that the simplification of $\sqrt{\sqrt{12}}$ is $\sqrt[4]{12}$.

To solve $\sqrt{\sqrt{2}}$, we will use the property of roots.

• Step 1: Recognize that $\sqrt{\sqrt{2}}$ involves two square roots.
• Step 2: Each square root can be expressed using exponents: $\sqrt{2} = 2^{1/2}$.
• Step 3: Therefore, $\sqrt{\sqrt{2}} = (2^{1/2})^{1/2}$.
• Step 4: Apply the formula for the root of a root: $(x^{a})^{b} = x^{ab}$.
• Step 5: For $(2^{1/2})^{1/2}$, this means we compute the product of the exponents: $(1/2) \times (1/2) = 1/4$.
• Step 6: The expression simplifies to $2^{1/4}$, which is written as $\sqrt[4]{2}$.

Therefore, $\sqrt{\sqrt{2}} = \sqrt[4]{2}$.

This corresponds to choice 2: $\sqrt[4]{2}$.

The solution to the problem is $\sqrt[4]{2}$.

To solve this problem, we'll observe the following process:

• Step 1: Recognize the expression $\sqrt[10]{\sqrt[10]{1}}$ involves nested roots.
• Step 2: Apply the formula for nested roots: $\sqrt[n]{\sqrt[m]{x}} = \sqrt[n \cdot m]{x}$.
• Step 3: Set $n = 10$ and $m = 10$, resulting in $\sqrt[10 \times 10]{1} = \sqrt[100]{1}$.
• Step 4: Simplify $\sqrt[100]{1}$. Any root of 1 is 1, as $1^k = 1$ for any positive rational number $k$.

Thus, the evaluation of the original expression $\sqrt[10]{\sqrt[10]{1}}$ equals 1.

Comparing this result to the provided choices:

• Choice 1 is $1$.
• Choice 2 is $\sqrt[100]{1}$, which is also 1.
• Choice 3 is $\sqrt{1} = 1$.
• Choice 4 states all answers are correct.

Therefore, choice 4 is correct: All answers are equivalent to the solution, being 1.

Thus, the correct selection is: All answers are correct.

To solve this 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.

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

• Step 1: Identify the inner and outer roots.
• Step 2: Apply the root of a root formula.
• Step 3: Calculate the combined root.

Now, let's work through each step:

Step 1: The expression given is $\sqrt[7]{\sqrt[4]{14}}$. The inner root is $\sqrt[4]{14}$, and the outer root is $\sqrt[7]{(\cdot)}$.

Step 2: Use the formula for the root of a root: $\sqrt[n]{\sqrt[m]{x}} = \sqrt[n \cdot m]{x}$.

Step 3: Plug in our values: $n = 7$ and $m = 4$. Thus, we have:

$\sqrt[7]{\sqrt[4]{14}} = \sqrt[7 \times 4]{14} = \sqrt[28]{14}$

Therefore, the simplified form of the given expression is $\sqrt[28]{14}$.

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

• Step 1: Use the formula for the nested radical: $\sqrt[10]{\sqrt[5]{100}} = \sqrt[10 \times 5]{100}$.
• Step 2: Calculate the product of the indices of the roots: $10 \times 5 = 50$.
• Step 3: Write the simplified expression: $\sqrt[50]{100}$.

Let's apply these steps to find the solution:

First, we apply the root of a root property:
$\sqrt[10]{\sqrt[5]{100}} = \sqrt[10 \times 5]{100}$

This simplifies to:
$\sqrt[50]{100}$

Therefore, the solution to the problem is $\sqrt[50]{100}$.

To solve $\sqrt[4]{\sqrt{6}}$, we'll simplify it using the property of roots:

According to the property $\sqrt[n]{\sqrt[m]{x}} = \sqrt[n \cdot m]{x}$, we are to multiply the indices of the roots.

Here, the indices are 4 (for the fourth root) and 2 (for the square root), hence $n = 4$ and $m = 2$.

Therefore, we calculate:

$4 \times 2 = 8$

This indicates that the expression can be rewritten as a single root of index 8, giving us:

$\sqrt[4]{\sqrt{6}} = \sqrt[8]{6}$

The answer is therefore $\sqrt[8]{6}$, which matches answer choice 4.

Let's solve the given problem by following these steps:

• Step 1: Recognize the expression $\sqrt[7]{\sqrt{2}}$. It involves two roots.
• Step 2: Rewrite each part using rational exponents. We have $\sqrt{2} = 2^{1/2}$.
• Step 3: Substitute back, giving $\sqrt[7]{2^{1/2}}$ or $(2^{1/2})^{1/7}$.
• Step 4: Use the properties of exponents: $(a^m)^n = a^{m \cdot n}$.
• Step 5: Calculate the exponent: $(1/2) \cdot (1/7) = 1/14$.
• Step 6: This gives us $2^{1/14}$, which is equal to $\sqrt[14]{2}$.

Thus, the simplified expression is $\sqrt[14]{2}$.

Therefore, the solution to the problem is $\sqrt[14]{2}$.

To solve this problem, we need to evaluate $\sqrt[3]{\sqrt{729}}$.

Step 1: Convert the expression into exponent form. We know $\sqrt{729} = 729^{\frac{1}{2}}$. Thus, $\sqrt[3]{\sqrt{729}} = (729^{\frac{1}{2}})^{\frac{1}{3}}$.

Step 2: Apply the formula for exponents $(a^m)^n = a^{m \cdot n}$. Thus, $(729^{\frac{1}{2}})^{\frac{1}{3}} = 729^{\frac{1}{2 \cdot 1/3}} = 729^{\frac{1}{6}}$.

Step 3: Find the base of 729 as a power of an integer. Observing, 729 = $3^6$ because $3^6 = 729$.

Step 4: Substitute the power of the base: $729^{\frac{1}{6}} = (3^6)^{\frac{1}{6}} = 3^{6 \cdot \frac{1}{6}} = 3^1 = 3$.

Therefore, the solution to the problem is $\sqrt[3]{\sqrt{729}} = 3$.

To solve this problem, let's follow these steps:

• Express the square root as a fractional exponent.
• Express the cube root as another fractional exponent.
• Multiply the exponents together using the rule $(a^m)^n = a^{m \times n}$.
• Recapture the result as a root expression.

Let's apply these steps:
Step 1: The square root of 144 can be expressed as $144^{1/2}$.
Step 2: We need the cube root of this expression, so we have $(144^{1/2})^{1/3}$.
Step 3: Using the property of exponents $(a^m)^n = a^{m \times n}$, we multiply the exponents: $(144^{1/2})^{1/3} = 144^{(1/2) \times (1/3)} = 144^{1/6}$.
Step 4: Re-express this as a root: Since $144^{1/6}$ is equivalent to the sixth root, we have $\sqrt[6]{144}$.

Therefore, the solution to the problem is $\sqrt[6]{144}$, which corresponds to choice 3.

To solve this problem, we'll convert the roots into exponents and simplify:

• Step 1: Express the nested radicals in terms of exponents.
• Step 2: Simplify by multiplying the exponents.
• Step 3: Compare the simplified result to the given choices.

Now, let's work through each step:
Step 1: The cube root of 16 can be written as $16^{\frac{1}{3}}$. Thus, our expression becomes $\sqrt[4]{16^{\frac{1}{3}}}$.
Step 2: Apply the fourth root, which is an exponent of $\frac{1}{4}$. This gives us $(16^{\frac{1}{3}})^{\frac{1}{4}} = 16^{\frac{1}{3} \cdot \frac{1}{4}} = 16^{\frac{1}{12}}$.
Step 3: From the original question, the expression simplifies to $16^{\frac{1}{12}}$, which is equivalent to $\sqrt[12]{16}$. Therefore, the choices that are correct are the ones that reflect this equivalence.

Therefore, the solution to the problem involves recognizing that both $16^{\frac{1}{12}}$ and $\sqrt[12]{16}$ represent the same value, and thus, answers a and b are correct.

To solve this problem, we'll express the nested roots using exponents and then simplify:

We start with the inner expression:

$\sqrt{64}$ is equivalent to $64^{1/2}$.

Next, apply the cube root:

$\sqrt[3]{64^{1/2}}$ is equivalent to $(64^{1/2})^{1/3}$.

Using properties of exponents, we simplify the expression:

$(64^{1/2})^{1/3} = 64^{(1/2) \cdot (1/3)} = 64^{1/6}$.

Now, evaluate $64^{1/6}$:

Since $64 = 2^6$, we have:

$64^{1/6} = (2^6)^{1/6} = 2^{6 \cdot (1/6)} = 2^{1} = 2$.

Therefore, the solution to the exercise $\sqrt[3]{\sqrt{64}}$ is $\mathbf{2}$.

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

• Step 1: Evaluate the innermost square root, $\sqrt{625}$.
• Step 2: Evaluate the square root of the result from Step 1.

Now, let's work through each step:

Step 1: Evaluate $\sqrt{625}$.
The square root of 625 is 25, since $25 \times 25 = 625$. Thus, $\sqrt{625} = 25$.

Step 2: Evaluate $\sqrt{25}$.
The square root of 25 is 5, since $5 \times 5 = 25$. Thus, $\sqrt{25} = 5$.

Therefore, the solution to the problem $\sqrt{\sqrt{625}}$ is 5.

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

• Step 1: Simplify the inner square root.
• Step 2: Apply the formula for nested roots.
• Step 3: Compute the fifth root.

Now, let's work through each step:

Step 1: Calculate the inner square root.
We have $\sqrt{1024}$. We know that $1024 = 2^{10}$, so $\sqrt{1024} = \sqrt{2^{10}}$.

Applying the property of roots, $\sqrt{2^{10}} = 2^{\frac{10}{2}} = 2^5 = 32$.

Step 2: Now, apply the fifth root to the result from step 1.
We need to find $\sqrt[5]{32}$.

Step 3: Simplify using the properties of exponents.
From $\sqrt[5]{32} = \sqrt[5]{2^5}$, we have $2^{\frac{5}{5}} = 2^1 = 2$.

Therefore, the solution to the problem is $2$.

To solve this problem, let's analyze and simplify the given expression $\sqrt[3]{\sqrt{36}}$.

• Step 1: Identify the root operations. We have a square root, $\sqrt{36}$, and a cube root, $\sqrt[3]{\ldots}$.

• Step 2: Use the formula for roots for a root of a root: $\sqrt[m]{\sqrt[n]{x}} = x^{\frac{1}{mn}}$.

• Step 3: Apply this formula to the problem. In this case, the first operation is a square root, which can be written as $36^{\frac{1}{2}}$, and the second operation is a cube root. Therefore, $\sqrt[3]{\sqrt{36}} = (36^{\frac{1}{2}})^{\frac{1}{3}}$.

• Step 4: Simplify using the power of a power rule, which allows us to multiply exponents: $(36^{\frac{1}{2}})^{\frac{1}{3}} = 36^{\frac{1}{2 \times 3}} = 36^{\frac{1}{6}}$.

Thus, the expression $\sqrt[3]{\sqrt{36}}$ simplifies to $36^{\frac{1}{6}}$.

To solve this problem, we'll proceed with the following steps:

• Step 1: Compute $\sqrt[3]{512}$.
  Given $512$, recognize that $512 = 2^9$ since $2^9 = 512$. Thus, we have:
• $\sqrt[3]{512} = \sqrt[3]{2^9} = 2^{9/3} = 2^3 = 8$.
• Step 2: Compute $\sqrt[3]{8}$.
  From Step 1, we found $\sqrt[3]{512} = 8$. Now find $\sqrt[3]{8}$:
• $\sqrt[3]{8} = \sqrt[3]{2^3} = 2^{3/3} = 2$.

Therefore, the solution to the expression $\sqrt[3]{\sqrt[3]{512}}$ is $2$.