Root of a Root: Using variables

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 $\sqrt{\sqrt{16 \cdot x^2}}$, 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 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 $\sqrt[3]{\sqrt{64 \cdot x^{12}}}$, 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 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 $3x$.

To simplify the given expression $\sqrt[4]{\sqrt[3]{49 \cdot x}}$, we will use the property of roots as fractional exponents.

• Step 1: Convert the cube root to a fractional exponent. We have $\sqrt[3]{49 \cdot x} = (49 \cdot x)^{1/3}$.
• Step 2: Apply the fourth root to the result, expressed as a fractional exponent. Thus, $\sqrt[4]{(49 \cdot x)^{1/3}} = ((49 \cdot x)^{1/3})^{1/4}$.
• Step 3: Combine the exponents by multiplying the fractions: $((49 \cdot x)^{1/3})^{1/4} = (49 \cdot x)^{(1/3) \times (1/4)} = (49 \cdot x)^{1/12}$.
• Step 4: Recognize that the result $(49 \cdot x)^{1/12}$ can be expressed as $\sqrt[12]{49 \cdot x}$.

Therefore, the solution to the problem is $\sqrt[12]{49x}$.

To solve $\sqrt[6]{\sqrt{x^{12}}}$, we will follow these steps:

• Step 1: Simplify the inner radical expression $\sqrt{x^{12}}$.
• Step 2: Use the property of roots, expressing $\sqrt{x^{12}}$ as a power of $x$.
• Step 3: Use the result from step 1 in the outer root expression.
• Step 4: Simplify the entire expression using exponent rules.

Now, let's perform each of these steps:

Step 1: Simplify $\sqrt{x^{12}}$.
$\sqrt{x^{12}} = x^{12/2} = x^6$.

Step 2: Simplify the outer expression $\sqrt[6]{x^6}$.
$\sqrt[6]{x^6} = (x^6)^{1/6}$.

Step 3: Apply the exponent rule $(a^m)^{n} = a^{m \times n}$.
$(x^6)^{1/6} = x^{6 \times 1/6} = x^1 = x$.

Therefore, the simplified expression is $\boxed{x}$.

Thus, the solution to $\sqrt[6]{\sqrt{x^{12}}}$ is $x$.

To solve the expression $\sqrt[3]{\sqrt{144x^3}}$, let's proceed step by step:

• Step 1: Express the inner square root using fractional exponents:
  $\sqrt{144x^3} = (144x^3)^{1/2}$
• Step 2: Apply the cube root to the expression obtained in Step 1:
  $\sqrt[3]{(144x^3)^{1/2}} = ((144x^3)^{1/2})^{1/3}$
• Step 3: Use the exponent rule $(a^m)^n = a^{m \cdot n}$:
  $((144x^3)^{1/2})^{1/3} = (144x^3)^{(1/2) \cdot (1/3)} = (144x^3)^{1/6}$
• Step 4: Separate the powers for 144 and $x^3$:
  $(144)^{1/6} \cdot (x^3)^{1/6} = \sqrt[6]{144} \cdot x^{3/6} = \sqrt[6]{144} \cdot x^{1/2} = \sqrt[6]{144} \cdot \sqrt{x}$

Therefore, the simplified expression is $\sqrt[6]{144} \cdot \sqrt{x}$, which corresponds to the third choice.

To solve the given problem, we'll follow these steps:

• Step 1: Simplify the innermost cube root $\sqrt[3]{512x^{27}}$.
• Step 2: Simplify the next cube root $\sqrt[3]{(\cdot)}$ from the result of step 1.

Let's go through each step:

Step 1: Consider the expression $\sqrt[3]{512x^{27}}$.
First, evaluate $\sqrt[3]{512}$. Since $512 = 8^3$, we have $\sqrt[3]{512} = 8$.
For $\sqrt[3]{x^{27}}$, use the property $\sqrt[n]{a^m} = a^{m/n}$:
$\sqrt[3]{x^{27}} = x^{27/3} = x^9$.
Thus, $\sqrt[3]{512x^{27}} = 8x^9$.

Step 2: Now, evaluate the outer cube root $\sqrt[3]{8x^9}$.
$\sqrt[3]{8} = 2$ since $8 = 2^3$.
For $\sqrt[3]{x^9}$, again use the rule $\sqrt[n]{a^m} = a^{m/n}$:
$\sqrt[3]{x^9} = x^{9/3} = x^3$.
Therefore, $\sqrt[3]{8x^9} = 2x^3$.

In conclusion, the simplified expression is $2x^3$.

Thus, the solution to the problem is $2x^3$, which corresponds to choice 3.

To solve the problem $\sqrt[5]{\sqrt{x^{20}}}$, we will convert the roots into fractional exponents and simplify:

The expression $\sqrt{x^{20}}$ can be represented as:

$(x^{20})^{\frac{1}{2}} = x^{20 \cdot \frac{1}{2}} = x^{10}$

Now, taking the fifth root of $x^{10}$, we have:

$(x^{10})^{\frac{1}{5}} = x^{10 \cdot \frac{1}{5}} = x^{2}$

Therefore, the original expression $\sqrt[5]{\sqrt{x^{20}}}$ simplifies to $x^2$.

To solve the problem, we'll simplify $\sqrt[10]{\sqrt{x^{20}}}$ using properties of exponents and roots:

• Step 1: Convert each part into exponent form. We have $\sqrt{x^{20}}$, which can be rewritten as $(x^{20})^{1/2}$.
• Step 2: Simplify the expression $(x^{20})^{1/2}$. Using the power of a power property, this becomes $x^{20 \cdot (1/2)} = x^{10}$.
• Step 3: Apply the 10th root to the simplified result: $\sqrt[10]{x^{10}}$, which can be written as $(x^{10})^{1/10}$.
• Step 4: Simplify $(x^{10})^{1/10}$ using the power of a power property again, we get $x^{(10 \cdot (1/10))} = x^1 = x$.

Therefore, the expression simplifies to $x$.

Conclusion: The solution to the problem is $x$.

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

• Step 1: Convert the expression under the roots to fractional exponents.
• Step 2: Simplify the fractional exponents.
• Step 3: Return the expression to root form if needed.

Now, let's work through each step:

Step 1: We have the expression $\sqrt[6]{\sqrt{25x^6}}$. Begin by simplifying the inner square root:

$\sqrt{25x^6} = (25x^6)^{1/2} = 25^{1/2} \cdot (x^6)^{1/2}$.

We know $25^{1/2} = 5$ and $(x^6)^{1/2} = x^{6 \cdot 1/2} = x^3$.

So, $\sqrt{25x^6} = 5x^3$.

Step 2: Apply the outer sixth root:

$\sqrt[6]{5x^3} = (5x^3)^{1/6} = 5^{1/6} \cdot (x^3)^{1/6} = 5^{1/6} \cdot x^{1/2}$.

Convert $5^{1/6}$ as $\sqrt[12]{25}$:

Since $5^{1/6} = (5^{1/2})^{1/3} = \sqrt{5}^{1/3} = \sqrt[12]{25}$,

we have $5^{1/6} = \sqrt[12]{25}$.

Therefore, the solution becomes:

$\sqrt{x} \cdot \sqrt[12]{25}$.

Therefore, the solution to the problem is $\sqrt{x}\cdot\sqrt[12]{25}$.