Root of a Root - Examples, Exercises and Solutions

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 $\sqrt{2}$.

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.

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 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 $\sqrt[4]{12}$.

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.

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.

To solve the problem $\sqrt[8]{\sqrt{x^8}}$, we'll simplify the expression using exponent rules:

• Step 1: Express the inner square root using exponents. We know $\sqrt{x^8} = (x^8)^{1/2} = x^{8 \cdot 1/2} = x^4$.
• Step 2: Express the entire expression with the 8th root as an exponent. We have $\sqrt[8]{x^4} = (x^4)^{1/8}$.
• Step 3: Simplify the expression, using $(x^a)^{b} = x^{a \cdot b}$. Therefore, $(x^4)^{1/8} = x^{4 \cdot 1/8} = x^{1/2}$.
• Step 4: Recognize $x^{1/2}$ is another way to write $\sqrt{x}$.

Thus, the expression simplifies to $\sqrt{x}$.

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.