Root of a Root - Examples, Exercises and Solutions

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 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 $\sqrt[15]{5}$.

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 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 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 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 the problem $\sqrt[3]{\sqrt{64 \cdot x^{12}}}$, follow these detailed steps:

• Step 1: Simplify the inner expression.
  The expression inside the radical is $64 \cdot x^{12}$.
• Step 2: Simplify the inner square root.
  

  First, we need to find $\sqrt{64 \cdot x^{12}}$.

  The square root of a product can be expressed as the product of the square roots: $\sqrt{64} \cdot \sqrt{x^{12}}$.

  Simplifying further, we find:

  • $\sqrt{64} = 8$, since $8^2 = 64$.
  • $\sqrt{x^{12}} = x^{6}$, because $(x^{6})^2 = x^{12}$.

  Thus, the inner square root becomes $8x^6$.

• Step 3: Simplify using the cube root.
  

  Next, apply the cube root to the result of the inner square root: $\sqrt[3]{8x^6}$.

  The cube root of a product can also be expressed as the product of the cube roots:

  • $\sqrt[3]{8} = 2$, since $2^3 = 8$.
  • $\sqrt[3]{x^6} = x^{6/3} = x^{2}$, because $(x^2)^3 = x^6$.

  Thus, the expression simplifies to $2x^2$.

Therefore, the solution to this problem is $2x^2$, which corresponds to choice 2 in the provided options.

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 the expression $\sqrt{\sqrt{16 \cdot x^2}}$, follow these steps:

• Step 1: Simplify the innermost root $\sqrt{16 \cdot x^2}$.
  Here, apply $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$. Thus, $\sqrt{16 \cdot x^2} = \sqrt{16} \cdot \sqrt{x^2}$.
• Step 2: Calculate each component:
  - $\sqrt{16} = 4$ because $16^{1/2} = 4$.
  - $\sqrt{x^2} = x$ assuming $x$ is non-negative.
• Step 3: Combine results from Step 2: $\sqrt{16} \cdot \sqrt{x^2} = 4 \cdot x = 4x$.
• Step 4: Simplify the outer square root: $\sqrt{4x}$.
  Applying $\sqrt{4x} = \sqrt{4} \cdot \sqrt{x}$, we have $\sqrt{4} = 2$.
  Thus, $\sqrt{4x} = 2 \cdot \sqrt{x} = 2\sqrt{x}$.

Therefore, the simplified form of $\sqrt{\sqrt{16 \cdot x^2}}$ is $2\sqrt{x}$. This corresponds to choice 1.

To solve $\sqrt{\sqrt{3x^2}}$, follow these steps:

• Step 1: Express the problem using exponentiation. The expression $\sqrt{3x^2}$ can be written as $(3x^2)^{\frac{1}{2}}$.
• Step 2: Take the square root of the first expression. This can be expressed as $((3x^2)^{\frac{1}{2}})^{\frac{1}{2}}$.
• Step 3: Use the property $(a^m)^n = a^{m \cdot n}$. Thus, $((3x^2)^{\frac{1}{2}})^{\frac{1}{2}} = (3x^2)^{\frac{1}{4}}$.
• Step 4: Simplify further using exponent rules: $(3x^2)^{\frac{1}{4}}$ becomes $(3^{\frac{1}{4}} \cdot (x^2)^{\frac{1}{4}})$, which simplifies to $\sqrt[4]{3} \cdot x^{\frac{1}{2}}$.
• Step 5: Recognize this as $\sqrt[4]{3} \cdot \sqrt{x}$, since $x^{\frac{1}{2}}$ is $\sqrt{x}$.

Therefore, the simplified form of the given expression is $\sqrt[4]{3} \cdot \sqrt{x}$.

To solve the expression $\sqrt[]{\sqrt{5x^4}}$, let's go step-by-step:

• Step 1: Simplify the inner expression $\sqrt{5x^4}$. Using the rule for square roots, we can rewrite $\sqrt{5x^4}$ as $(5x^4)^{1/2}$. This expression can be further simplified to $5^{1/2} \cdot (x^4)^{1/2} = \sqrt{5} \cdot x^2$.
• Step 2: Take the square root of the simplified expression. This means we apply another square root to $\sqrt{5} \cdot x^2$, resulting in $(\sqrt{5} \cdot x^2)^{1/2} = (\sqrt{5})^{1/2} \cdot (x^2)^{1/2}$.
• Step 3: Simplify each component: $\sqrt[4]{5} \cdot x$. We find that $(\sqrt{5})^{1/2}$ simplifies to $\sqrt[4]{5}$ and $(x^2)^{1/2}$ to $x$.

Therefore, the simplified expression is $\sqrt[4]{5} \cdot x$.

To solve the problem $\sqrt{\sqrt{81 \cdot x^4}}$, we need to simplify this expression using properties of exponents and square roots.

• Step 1: Simplify the inner square root
  The expression inside the first square root is $81 \cdot x^4$. We can rewrite this using exponents:
  $81 = 9^2$ and $x^4 = (x^2)^2$. Thus, $81 \cdot x^4 = (9x^2)^2$.
• Step 2: Apply the inner square root
  Taking the square root of $(9x^2)^2$ gives us:
  $\sqrt{(9x^2)^2} = 9x^2$, because $\sqrt{a^2} = a$ where $a$ is a non-negative real number.
• Step 3: Simplify the outer square root
  Now, we take the square root of the result from the inner root:
  $\sqrt{9x^2} = \sqrt{9} \cdot \sqrt{x^2} = 3 \cdot x = 3x$, since $\sqrt{x^2} = x$ given $x$ is non-negative.

Therefore, the solution to the problem is $3x$.