Root of a Root: Number of terms

To find the value of $\sqrt{\sqrt{49}} \cdot \sqrt{\sqrt{16}}$, we will follow these steps:

• Step 1: Simplify $\sqrt{\sqrt{49}}$.
• Step 2: Simplify $\sqrt{\sqrt{16}}$.
• Step 3: Multiply the simplified results together.

Step 1: $\sqrt{\sqrt{49}}$
- Calculate $\sqrt{49} = 7$.
- Therefore, $\sqrt{\sqrt{49}} = \sqrt{7}$.

Step 2: $\sqrt{\sqrt{16}}$
- Calculate $\sqrt{16} = 4$.
- Therefore, $\sqrt{\sqrt{16}} = \sqrt{4} = 2$.

Step 3: Multiply the simplified results:
- Multiply $\sqrt{7}$ by $2$.
- The product is $2 \cdot \sqrt{7} = 2\sqrt{7}$.

Therefore, the value of $\sqrt{\sqrt{49}} \cdot \sqrt{\sqrt{16}}$ is $\mathbf{2\sqrt{7}}$.

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

• Step 1: Simplify each term individually.
• Step 2: Multiply the simplified terms together.
• Step 3: Compare with choices if necessary.

Let's begin:

Step 1: Simplify each term:

The expression is $\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{4}}$.

- Simplifying $\sqrt{\sqrt{2}}$: A root of a root involves multiplying the indices. We have $\sqrt[2]{\sqrt[2]{2}}$, which becomes $\sqrt[4]{2}$.

- Simplifying $\sqrt{\sqrt{4}}$: Note that $\sqrt{4} = 2$, so $\sqrt{2} = \sqrt{2}$.

Conclusively, $\sqrt{\sqrt{4}} = \sqrt{2}$.

Step 2: Multiply the simplified terms:

Now, multiply $\sqrt[4]{2} \times \sqrt{2}$:

$\sqrt[4]{2} \cdot \sqrt{2} = \sqrt[4]{2 \times 2^{1/2}} = \sqrt[4]{2^{3/2}} = \sqrt[4]{8}$.

Therefore, our simplified expression is $\sqrt[4]{8}$.

Step 3: Compare with answer choices:

The correct choice is $\sqrt[4]{8}$, matching choice 3.

Therefore, the solution to the problem is $\sqrt[4]{8}$.

To solve the problem $\sqrt{\sqrt{16}}\cdot\sqrt{\sqrt{8}}$, we will follow these steps:

• Step 1: Simplify each nested square root expression.
• Step 2: Multiply the simplified expressions of the roots.

Step 1: Evaluate $\sqrt{\sqrt{16}}$.
Since 16 can be expressed as $2^4$, we have: $\sqrt{\sqrt{16}} = \left(16^{1/2}\right)^{1/2} = 16^{1/4} = \left(2^4\right)^{1/4} = 2^{4/4} = 2^{1} = 2$

Evaluate $\sqrt{\sqrt{8}}$.
Since 8 can be expressed as $2^3$, we have: $\sqrt{\sqrt{8}} = \left(8^{1/2}\right)^{1/2} = 8^{1/4} = \left(2^3\right)^{1/4} = 2^{3/4}$

Step 2: Multiply these simplified expressions together: $2 \cdot 2^{3/4} = 2^{1} \cdot 2^{3/4} = 2^{1 + 3/4} = 2^{7/4}$

Finally, converting back to radical form: $2^{7/4} = \left(2^7\right)^{1/4} = \sqrt[4]{2^7} = \sqrt[4]{128}$

Thus, the solution to the problem is $\sqrt[4]{128}$, which corresponds to answer choice 3.

To solve the expression $\sqrt{\sqrt{4}}\cdot\sqrt{\sqrt{2}}$, we will use properties of exponents and roots.

First, let's simplify each part:

• $\sqrt{\sqrt{4}}$:

We know $\sqrt{4} = 2$. Therefore, $\sqrt{\sqrt{4}}$ can be rewritten as $\sqrt{2}$, because $\sqrt{4} = 2$ and further taking square root gives $2^{1/2}$.

• $\sqrt{\sqrt{2}}$:

This expression is equivalent to $(2^{1/2})^{1/2}$. Using the property $(a^{m})^{n} = a^{m \cdot n}$, we have:

$(2^{1/2})^{1/2} = 2^{1/4}$.

Now, the original expression simplifies to:

$\sqrt{2} \cdot 2^{1/4}$

This product is expressed as:

$2^{1/2} \cdot 2^{1/4}$. When multiplying like bases, add the exponents:

$2^{1/2 + 1/4} = 2^{2/4 + 1/4} = 2^{3/4}$

Thus, the final expression is:

$2^{1/4}\sqrt{2}$.

Comparing this to the choices provided, the correct answer is:

$2^{1/4}\sqrt{2}$ (Choice 3).

Therefore, the solution to the problem is $\boxed{2^{\frac{1}{4}}\sqrt{2}}$.

To solve the problem $\sqrt{25} \cdot \sqrt[3]{\sqrt{25}}$, follow these steps:

• First, express each root in terms of exponents:
• $\sqrt{25} = 25^{1/2}$
• $\sqrt[3]{\sqrt{25}} = \sqrt[3]{25^{1/2}}$
• Using the law of exponents $(a^m)^n = a^{m \cdot n}$, simplify $\sqrt[3]{25^{1/2}}$ as follows:
• $(25^{1/2})^{1/3} = 25^{(1/2) \cdot (1/3)} = 25^{1/6}$
• Now, multiply the two expressions:
• $25^{1/2} \cdot 25^{1/6} = 25^{(1/2 + 1/6)}$
• Calculate the sum of the exponents: $\frac{1}{2} + \frac{1}{6} = \frac{3}{6} + \frac{1}{6} = \frac{4}{6} = \frac{2}{3}$
• This gives: $25^{2/3}$
• Recognize $25 = 5^2$, thus: $(5^2)^{2/3} = 5^{2 \cdot (2/3)} = 5^{4/3}$
• Convert mixed fraction: $5^{4/3} = 5^{1 + 1/3} = 5^{1\frac{1}{3}}$

Therefore, the product $\sqrt{25} \cdot \sqrt[3]{\sqrt{25}}$ equals $\mathbf{5^{1\frac{1}{3}}}$.

To solve this problem, we will simplify the expression $\sqrt[3]{\sqrt{16}} \cdot \sqrt[]{\sqrt{16}}$ using the rules for exponents and roots.

First, consider the inner square root $\sqrt{16}$. We know that:
$\sqrt{16} = 16^{1/2}$

Next, we address the cube root term $\sqrt[3]{\sqrt{16}}$. Express $\sqrt{16}$ as $16^{1/2}$, then:

• $\sqrt[3]{\sqrt{16}} = \sqrt[3]{16^{1/2}}$ Convert to exponents: $\sqrt[3]{16^{1/2}} = (16^{1/2})^{1/3} = 16^{(1/2) \cdot (1/3)} = 16^{1/6}$
• The other term is $\sqrt{\sqrt{16}} = \sqrt{16^{1/2}}$ Convert to exponents: $\sqrt{16^{1/2}} = (16^{1/2})^{1/2} = 16^{(1/2) \cdot (1/2)} = 16^{1/4}$

Now, multiply these results:
$16^{1/6} \cdot 16^{1/4}$

Using the product rule for exponents $(a^m \cdot a^n = a^{m+n})$, combine the exponents:
$16^{1/6 + 1/4}$

Find the common denominator to add the fractions:

• $1/6 = 2/12$
• $1/4 = 3/12$ $1/6 + 1/4 = 2/12 + 3/12 = 5/12$

Thus, the expression becomes:
$16^{5/12}$

Therefore, the simplified expression is $16^{\frac{5}{12}}$.

To solve the problem $\sqrt[3]{\sqrt{25}} \cdot \sqrt[3]{\sqrt{64}}$, we will work through it step by step:

Step 1: Simplify the inner square roots.

• $\sqrt{25}$ simplifies to $5$ because $5 \times 5 = 25$.
• $\sqrt{64}$ simplifies to $8$ because $8 \times 8 = 64$.

Step 2: Evaluate the cube roots.

• $\sqrt[3]{5}$ remains as $5^{1/3}$.
• $\sqrt[3]{8}$ evaluates to $2$ because $2 \times 2 \times 2 = 8$.

Step 3: Multiply the results of the cube roots.

• $5^{1/3} \cdot 2 = 2 \sqrt[3]{5}$.

Thus, the simplified expression is $2 \sqrt[3]{5}$.

Therefore, the solution to the problem is $2\sqrt[3]{5}$.

To solve the problem $\sqrt[5]{\sqrt{3}} \cdot \sqrt[5]{\sqrt{3}} =$, we follow these steps:

Step 1: Express each root using exponents.
$\sqrt[5]{\sqrt{3}}$ can be rewritten as $(3^{1/2})^{1/5}$, which simplifies to $3^{1/10}$ using the law $(a^m)^n = a^{m \cdot n}$.

Step 2: Multiply the expressions.
We have $(3^{1/10}) \cdot (3^{1/10})$. According to the laws of exponents, $a^m \cdot a^n = a^{m+n}$. Thus, the expression becomes $3^{1/10 + 1/10} = 3^{2/10} = 3^{1/5}$.

Step 3: Convert back to a root, if necessary.
The expression $3^{1/5}$ corresponds to $\sqrt[5]{3}$.

Therefore, the expression $\sqrt[5]{\sqrt{3}} \cdot \sqrt[5]{\sqrt{3}}$ simplifies to $3^{1/5}$, which is equivalent to $\sqrt[5]{9}$.

To match with the given choices, observe that $3^{1/5}$ can also be expressed as $\sqrt[10]{9}$ because $3^{1/5} = (3^2)^{1/10}$, which equals to $\sqrt[10]{9}$.

The correct answer is choice 4, $\sqrt[10]{9}$.

To solve $\sqrt[3]{\sqrt{3}} \cdot \sqrt[3]{\sqrt{4}}$, we will convert these expressions into powers:

Step 1: Express each root as a power:
$\sqrt[3]{\sqrt{3}} = (\sqrt{3})^{1/3} = 3^{1/6}$
$\sqrt[3]{\sqrt{4}} = (\sqrt{4})^{1/3} = 4^{1/6}$

Step 2: Multiply the expressions using the property of exponents:
$3^{1/6} \cdot 4^{1/6} = (3 \cdot 4)^{1/6} = 12^{1/6}$

Therefore, the simplified expression is $\sqrt[6]{12}$.

To simplify the expression $\sqrt[5]{\sqrt{3}} \cdot \sqrt[6]{\sqrt{3}}$, follow these steps:

• Step 1: Represent $\sqrt{3}$ as $3^{1/2}$ because $\sqrt{3} = 3^{1/2}$.
• Step 2: Express $\sqrt[5]{\sqrt{3}}$ in exponential form: $\sqrt[5]{\sqrt{3}} = (\sqrt{3})^{1/5} = (3^{1/2})^{1/5} = 3^{(1/2) \times (1/5)} = 3^{1/10}$.
• Step 3: Express $\sqrt[6]{\sqrt{3}}$ in exponential form: $\sqrt[6]{\sqrt{3}} = (\sqrt{3})^{1/6} = (3^{1/2})^{1/6} = 3^{(1/2) \times (1/6)} = 3^{1/12}$.
• Step 4: Multiply the two expressions using properties of exponents: $3^{1/10} \cdot 3^{1/12} = 3^{(1/10 + 1/12)}$.

Therefore, the simplified expression is $3^{\frac{1}{10}+\frac{1}{12}}$.

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

• Step 1: Simplify $\sqrt{64}$.
• Step 2: Simplify with $\sqrt[6]{}$.
• Step 3: Simplify $\sqrt[3]{16}$.
• Step 4: Simplify with $\sqrt[4]{}$.
• Step 5: Combine the simplified results.

Now, let's work through each step:

Step 1: Simplify $\sqrt{64}$.
Since $64 = 8^2$, we have $\sqrt{64} = 8$.

Step 2: Simplify $\sqrt[6]{8}$ using exponent rules:
$\sqrt[6]{8} = 8^{1/6}$.

Step 3: Simplify $\sqrt[3]{16}$.
Since $16 = 2^4$, then $\sqrt[3]{16} = (2^4)^{1/3} = 2^{4/3}$.

Step 4: Simplify $\sqrt[4]{2^{4/3}}$ using exponent rules:
We have $\sqrt[4]{2^{4/3}} = (2^{4/3})^{1/4} = 2^{4/12} = 2^{1/3}$.

Step 5: Combine simplified results:
We have $8^{1/6} \cdot 2^{1/3} = (2^3)^{1/6} \cdot 2^{1/3} = 2^{3/6} \cdot 2^{1/3} = 2^{1/2} \cdot 2^{1/3} = 2^{(1/2 + 1/3)} = 2^{(3/6 + 2/6)} = 2^{5/6}.$

Convert to a common root form:
The result is equivalent to $2^{5/6} = \sqrt[12]{2^{10}} = \sqrt[12]{1024}$.

Therefore, the solution to the problem is $\sqrt[12]{1024}$.

Let's solve the problem step-by-step.

• Step 1: Simplify $\sqrt[3]{\sqrt{64}}$.
• Step 2: Simplify $\sqrt[3]{64}$.
• Step 3: Multiply the results of Step 1 and Step 2.

Step 1: Consider $\sqrt[3]{\sqrt{64}}$.

We can write $\sqrt{64}$ as $64^{1/2}$. Thus, $\sqrt[3]{\sqrt{64}} = \sqrt[3]{64^{1/2}}$.

Using the property $\sqrt[n]{a^m} = a^{m/n}$, we have $(64^{1/2})^{1/3} = 64^{1/6}$.

Step 2: Simplify $\sqrt[3]{64}$.

The cube root of a number $b$ is expressed as $b^{1/3}$. Therefore, $\sqrt[3]{64} = 64^{1/3}$.

Step 3: Multiply the two results.

We now compute $64^{1/6} \cdot 64^{1/3}$.

Using the property of exponents, $a^m \cdot a^n = a^{m+n}$, thus $64^{1/6} \cdot 64^{1/3} = 64^{(1/6 + 1/3)} = 64^{(1/6 + 2/6)} = 64^{3/6} = 64^{1/2}$.

Finally, $64^{1/2}$ is simply $\sqrt{64}$, which equals $8$.

Therefore, the solution to the problem is $8$, which corresponds to choice (3).