Rules of Roots Combined: Using multiple rules

To solve this problem, we will systematically apply the properties of square roots and perform the arithmetic operations:

• Step 1: Calculate $\sqrt{100}$.
• Step 2: Calculate the individual square roots $\sqrt{25}$ and $\sqrt{4}$, and then multiply them.
• Step 3: Divide the result from Step 1 by the product of Step 2.

Now, let's work through each step:

Step 1: The square root of 100 is 10, since $10 \times 10 = 100$. Therefore, $\sqrt{100} = 10$.

Step 2: Calculate $\sqrt{25}$ and $\sqrt{4}$. We know $\sqrt{25} = 5$ because $5 \times 5 = 25$, and $\sqrt{4} = 2$ because $2 \times 2 = 4$. Thus, the product is $\sqrt{25} \cdot \sqrt{4} = 5 \cdot 2 = 10$.

Step 3: Divide the result from Step 1 by the product from Step 2: $\frac{\sqrt{100}}{\sqrt{25} \cdot \sqrt{4}} = \frac{10}{10}$.

Therefore, the simplified expression is $1$.

As a result, the answer to the problem is $1$.

In order to simplify the given expression, we will use the following two laws of exponents:

a. Law of exponents for multiplication of terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

b. Law of exponents for division of terms with identical bases:

$\frac{a^m}{a^n}=a^{m-n}$

Let's solve the given expression:

$\frac{2^3\cdot2^4}{2^5}=$First, since in the numerator we have multiplication of terms with identical bases, we'll use the law of exponents mentioned in a:

$\frac{2^3\cdot2^4}{2^5}= \\ \frac{2^{3+4}}{2^5}=\\ \frac{2^{7}}{2^5}=\\$We'll continue, since we have division of terms with identical bases, we'll use the law of exponents mentioned in b:

$\frac{2^{7}}{2^5}=\\ 2^{7-5}=\\ 2^2=\\ \boxed{4}$

Let's summarize the simplification of the given expression:

$\frac{2^3\cdot2^4}{2^5}= \\ \frac{2^{7}}{2^5}=\\ 2^2=\\ \boxed{4}$

Therefore, the correct answer is answer d.

Introduction:

We will address the following three laws of exponents:

a. Definition of root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

b. The law of exponents for exponents applied to multiplication of terms in parentheses:

$(a\cdot b)^n=a^n\cdot b^n$

c. The law of exponents for exponents applied to division of terms in parentheses:

$(\frac{a}{b})^n=\frac{a^n}{b^n}$

Note:

(1). By combining the two laws of exponents mentioned in a (in the first and third steps later) and b (in the second step later), we can obtain a new rule:

$\sqrt[n]{a\cdot b}=\\ (a\cdot b)^{\frac{1}{n}}=\\ a^{\frac{1}{n}}\cdot b^{\frac{1}{n}}=\\ \sqrt[n]{a}\cdot \sqrt[n]{ b}\\ \downarrow\\ \boxed{\sqrt[n]{a\cdot b}=\sqrt[n]{a}\cdot \sqrt[n]{ b}}$

And specifically for the fourth root we get:

$\boxed{ \sqrt{a\cdot b}=\sqrt{a}\cdot \sqrt{ b}}$

(2). Similarly, note that by combining the two laws of exponents mentioned in a (in the first and third steps later) and c (in the second step later), we can obtain another new rule:

$\sqrt[n]{\frac{a}{b}}=\\ (\frac{a}{b})^{\frac{1}{n}}=\\ \frac{a^{\frac{1}{n}}}{ b^{\frac{1}{n}}}=\\ \frac{\sqrt[n]{a}}{ \sqrt[n]{ b}}\\ \downarrow\\ \boxed{ \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{ \sqrt[n]{ b}}}$

And specifically for the fourth root we get:

$\boxed{ \sqrt{\frac{a}{ b}}=\frac{\sqrt{a}}{\sqrt{ b}} }$

Therefore, in solving the problem, meaning - in simplifying the given expression, we will use the two new rules we received in the introduction:

(1).

$\boxed{ \sqrt{a\cdot b}=\sqrt{a}\cdot \sqrt{ b}}$

(2).

$\boxed{ \sqrt{\frac{a}{ b}}=\frac{\sqrt{a}}{\sqrt{ b}} }$

We'll start by simplifying the expression in the numerator using the rule we received in the introduction (1) (but in the opposite direction, meaning we'll insert the multiplication of roots as a multiplication of terms under the same root) Then we'll perform the multiplication under the root in the numerator:

$\frac{\sqrt{20}\cdot\sqrt{4}}{\sqrt{5}}= \\ \frac{\sqrt{20\cdot4}}{\sqrt{5}}= \\ \frac{\sqrt{80}}{\sqrt{5}}= \\$We'll continue and simplify the fraction, using the rule we received in the introduction (2) (but in the opposite direction, meaning we'll insert the division of roots as a division of terms under the same root) Then we'll reduce the fraction under the root:

$\frac{\sqrt{80}}{\sqrt{5}}= \\ \\ \sqrt{\frac{80}{5}}=\\ \sqrt{16}=\\ \boxed{4}$

In the final stage, after reducing the fraction under the root, we used the known fourth root of the number 16.

Let's summarize the simplification process of the expression in the problem:

$\frac{\sqrt{20}\cdot\sqrt{4}}{\sqrt{5}}= \\ \frac{\sqrt{80}}{\sqrt{5}}= \\ \sqrt{16}=\\ \boxed{4}$

Therefore, the correct answer is answer B.

Let's begin the solution by applying the product property of square roots:

Combine the square roots in the numerator:

$\sqrt{35} \cdot \sqrt{20} = \sqrt{35 \cdot 20}$

Calculate $35 \cdot 20 = 700$, so:

$\sqrt{35} \cdot \sqrt{20} = \sqrt{700}$

Now, divide this square root by the square root in the denominator using the quotient property:

$\frac{\sqrt{700}}{\sqrt{7}} = \sqrt{\frac{700}{7}}$

Simplify the fraction inside the square root:

$\frac{700}{7} = 100$

Thus, the expression becomes:

$\sqrt{100} = 10$

Therefore, the solution to the expression $\frac{\sqrt{35} \cdot \sqrt{20}}{\sqrt{7}}$ is $10$.

The correct answer choice is:

$10$

To solve the expression $\sqrt[3]{\sqrt{64}}\cdot\sqrt{64}$, we follow these steps:

• Step 1: Express $\sqrt{64}$ as a power:
  Since $\sqrt{64} = 64^{1/2}$ and $64 = 2^6$, substituting gives $(2^6)^{1/2} = 2^{6 \cdot 1/2} = 2^3 = 8$.
• Step 2: Express $\sqrt[3]{\sqrt{64}}$ as a power:
  Since from Step 1, $\sqrt{64} = 2^3 = 8$, then $\sqrt[3]{\sqrt{64}} = \sqrt[3]{8}$.
  Now, $\sqrt[3]{8} = 8^{1/3}$ and $8 = 2^3$, so $(2^3)^{1/3} = 2^{3 \cdot 1/3} = 2^1 = 2$.
• Step 3: Multiply the simplified expressions:
  We now have $\sqrt[3]{\sqrt{64}} = 2$ and $\sqrt{64} = 8$.
  Thus, $\sqrt[3]{\sqrt{64}} \cdot \sqrt{64} = 2 \cdot 8 = 16$.

Therefore, the solution to the problem is $16$.

To simplify the given expression, we will use two laws of exponents:

A. Definition of the root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

B. Law of exponents for an exponent on an exponent:

$(a^m)^n=a^{m\cdot n}$

Let's begin simplifying the given expression:

$\sqrt[4]{\sqrt[3]{3}}= \\$We will use the law of exponents shown in A and first convert the roots in the expression to exponents, we will do this in two steps - in the first step we will convert the inner root in the expression and in the next step we will convert the outer root:

$\sqrt[4]{\sqrt[3]{3}}= \\ \sqrt[4]{3^{\frac{1}{3}}}= \\ (3^{\frac{1}{3}})^{\frac{1}{4}}=$We continue and use the law of exponents shown in B, then we will multiply the exponents:

$(3^{\frac{1}{3}})^{\frac{1}{4}}= \\ 3^{\frac{1}{3}\cdot\frac{1}{4}}=\\ 3^{\frac{1\cdot1}{3\cdot4}}=\\ \boxed{3^{\frac{1}{12}}}=\\ \boxed{\sqrt[12]{3}}$ In the final step we return to writing the root, that is - back, using the law of exponents shown in A (in the opposite direction),

Let's summarize the simplification of the given expression:

$\sqrt[4]{\sqrt[3]{3}}= \\ (3^{\frac{1}{3}})^{\frac{1}{4}}= \\ \boxed{3^{\frac{1}{12}}}=\\ \boxed{\sqrt[12]{3}}$Therefore, note that the correct answer (most) is answer D.

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

• Step 1: Simplify each square root:

$\sqrt{4} = 2, \quad \sqrt{9} = 3, \quad \sqrt{16} = 4$

• Step 2: Calculate the expression in the numerator:

The expression in the numerator is $\sqrt{4} \cdot \sqrt{9}$. Substituting the simplified values, we have:
$2 \cdot 3 = 6$

• Step 3: Compute the division with the denominator:

Now, divide the result from Step 2 by the simplified denominator:
$\frac{6}{4} = \frac{3}{2}$

Thus, the value of the expression is $\frac{3}{2}$.

Therefore, the solution to the problem is $\frac{3}{2}$.

Introduction:

We will address the following three laws of exponents:

a. Definition of root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

b. The law of exponents for exponents applied to multiplication of terms in parentheses:

$(a\cdot b)^n=a^n\cdot b^n$

c. The law of exponents for exponents applied to division of terms in parentheses:

$(\frac{a}{b})^n=\frac{a^n}{b^n}$

Note:

d. By combining the two laws of exponents mentioned in a' (in the first and third steps later) and b' (in the second step later), we can obtain a new rule:

$\sqrt[n]{a\cdot b}=\\ (a\cdot b)^{\frac{1}{n}}=\\ a^{\frac{1}{n}}\cdot b^{\frac{1}{n}}=\\ \sqrt[n]{a}\cdot \sqrt[n]{ b}\\ \downarrow\\ \boxed{\sqrt[n]{a\cdot b}=\sqrt[n]{a}\cdot \sqrt[n]{ b}}$

And specifically for the fourth root we get:

$\boxed{ \sqrt{a\cdot b}=\sqrt{a}\cdot \sqrt{ b}}$

e. Similarly, note that by combining the two laws of exponents mentioned in a' (in the first and third steps later) and c' (in the second step later), we can obtain another new rule:

$\sqrt[n]{\frac{a}{b}}=\\ (\frac{a}{b})^{\frac{1}{n}}=\\ \frac{a^{\frac{1}{n}}}{ b^{\frac{1}{n}}}=\\ \frac{\sqrt[n]{a}}{ \sqrt[n]{ b}}\\ \downarrow\\ \boxed{ \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{ \sqrt[n]{ b}}}$

And specifically for the fourth root we get:

$\boxed{ \sqrt{\frac{a}{ b}}=\frac{\sqrt{a}}{\sqrt{ b}} }$

Therefore, in solving the problem, that is - in simplifying the given expression, we will use the two new rules we received in the introduction:

(1).

$\boxed{ \sqrt{a\cdot b}=\sqrt{a}\cdot \sqrt{ b}}$$$(2).

$\boxed{ \sqrt{\frac{a}{ b}}=\frac{\sqrt{a}}{\sqrt{ b}} }$

We'll start by simplifying the expression in the numerator using the rule we received in the introduction (1) (but in the opposite direction, meaning we'll insert the multiplication of roots as a multiplication of terms under the same root) then we'll perform the multiplication under the root in the numerator:

$\frac{\sqrt{70}\cdot\sqrt{10}}{\sqrt{7}}= \\ \frac{\sqrt{70\cdot10}}{\sqrt{7}}= \\ \frac{\sqrt{700}}{\sqrt{7}}= \\$We'll continue and simplify the fraction, using the rule we received in the introduction (2) (but in the opposite direction, meaning we'll insert the division of roots as a division of terms under the same root) then we'll reduce the fraction under the root:

$\frac{\sqrt{700}}{\sqrt{7}}= \\ \sqrt{\frac{700}{7}}=\\ \sqrt{100}=\\ \boxed{10}$

In the final stage, after reducing the fraction under the root, we used the known fourth root of the number 100.

Let's summarize the process of simplifying the expression in the problem:

$\frac{\sqrt{70}\cdot\sqrt{10}}{\sqrt{7}}= \\ \frac{\sqrt{700}}{\sqrt{7}}= \\ \sqrt{100}=\\ \boxed{10}$

Therefore, the correct answer is answer a'.

Let's proceed to simplify the expression:

• First, evaluate the numerator: $\sqrt{2} \cdot \sqrt{9} \cdot \sqrt{2}$. Using the property $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$, we simplify it: $\sqrt{2 \cdot 9 \cdot 2} = \sqrt{36}$.
• $\sqrt{36}$ simplifies to 6, as 36 is a perfect square.
• Next, evaluate the denominator $\sqrt{3} \cdot \sqrt{4}$:
• $\sqrt{3} \cdot \sqrt{4}$ also applies the property $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$, simplifying to $\sqrt{12}$.
• Since 12 is $4 \times 3$, and $\sqrt{4} = 2$, $\sqrt{12}$ simplifies to $2\sqrt{3}$.
• Now, the original expression becomes $\frac{6}{2\sqrt{3}}$.
• Simplify $\frac{6}{2}$ to get $\frac{6}{2} = 3$.
• The entire expression now is $\frac{3}{\sqrt{3}}$.
• To rationalize the expression $\frac{3}{\sqrt{3}}$, multiply both the numerator and the denominator by $\sqrt{3}$:
• This becomes $\frac{3 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{3\sqrt{3}}{3} = \sqrt{3}$

Therefore, the solution to the problem is $\sqrt{3}$.

To solve the expression $\sqrt{\frac{36}{144}} \cdot \sqrt{\sqrt{16}}$, follow these steps:

• Simplify $\sqrt{\frac{36}{144}}$:
  - Evaluate the fraction: $\frac{36}{144} = \frac{1}{4}$.
  - Take the square root: $\sqrt{\frac{1}{4}} = \frac{1}{2}$ because $\sqrt{1} = 1$ and $\sqrt{4} = 2$.
• Simplify $\sqrt{\sqrt{16}}$:
  - First evaluate the inner square root: $\sqrt{16} = 4$ since $4^2 = 16$.
  - Then take the square root of the result: $\sqrt{4} = 2$ since $2^2 = 4$.
• Multiply the results from both parts:
  - Multiply the simplified results: $\frac{1}{2} \cdot 2 = 1$.

Therefore, the solution to the expression is $1$.

To solve the problem $\frac{\sqrt{10} \cdot \sqrt{30}}{\sqrt{100}}$, we'll use the rules of square roots, specifically the multiplication and division properties.

Start by simplifying the numerator using the multiplication property of square roots:

• $\sqrt{10} \cdot \sqrt{30} = \sqrt{10 \times 30} = \sqrt{300}$

Next, we simplify the entire fraction using the division property of square roots:

• $\frac{\sqrt{300}}{\sqrt{100}} = \sqrt{\frac{300}{100}} = \sqrt{3}$

Thus, the simplified form of the expression $\frac{\sqrt{10} \cdot \sqrt{30}}{\sqrt{100}}$ is $\sqrt{3}$.

Therefore, the solution to the problem is $\sqrt{3}$.

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

• Step 1: Calculate the cube root of 64
• Step 2: Simplify the fraction $\frac{16}{\sqrt[3]{64}}$
• Step 3: Simplify $\sqrt{\text{result from step 2}}$

Let's proceed with each step:
Step 1: The cube root of 64 is calculated as follows:
$\sqrt[3]{64} = 4$. This is because $4^3 = 64$.

Step 2: Now, simplify the fraction $\frac{16}{4}$:
$\frac{16}{4} = 4$.

Step 3: Finally, take the square root of the result from step 2:
$\sqrt{4} = 2$.

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

To solve this problem, let's evaluate the expression $\sqrt{\sqrt{\frac{100}{25}}} \cdot \sqrt{\sqrt{25}}$ step-by-step.

Step 1: Simplify $\sqrt{\frac{100}{25}}$.
We calculate $\frac{100}{25}$, which simplifies to 4. Therefore, $\sqrt{\frac{100}{25}} = \sqrt{4} = 2$.

Step 2: Now find $\sqrt{\sqrt{4}}$.
$\sqrt{4} = 2$, so $\sqrt{2}$ remains as it is.

Step 3: Simplify $\sqrt{\sqrt{25}}$.
Note that $\sqrt{25} = 5$. Therefore, $\sqrt{\sqrt{25}} = \sqrt{5}$.

Step 4: Combine the results:
The expression simplifies to $\sqrt{2} \cdot \sqrt{5}$, which is equal to $\sqrt{2 \cdot 5} = \sqrt{10}$.

Thus, the final result of the expression is $\sqrt{10}$.

The correct choice from the options provided is: $\sqrt{10}$.

To solve this problem, we'll use the properties of square roots:

• Step 1: Apply the multiplication property of square roots in the numerator:
  $\sqrt{2} \cdot \sqrt{6} \cdot \sqrt{12} = \sqrt{2 \cdot 6 \cdot 12}$
• Step 2: Calculate the product under the square root:
  $2 \cdot 6 \cdot 12 = 144$
• Step 3: Combine the expression:
  $\frac{\sqrt{144}}{\sqrt{16}}$
• Step 4: Simplify the square roots:
  $\sqrt{144} = 12$ and $\sqrt{16} = 4$
• Step 5: Use the properties of the quotient of square roots:
  $\frac{12}{4} = 3$

Thus, the final simplified expression is $\mathbf{3}$.

To solve the problem of finding $\sqrt[7]{\sqrt{5}} \cdot \sqrt[14]{\sqrt{5}}$, we will use properties of exponents. Here's how to proceed:

• Step 1: Express each component using exponent notation.
  $\sqrt[7]{\sqrt{5}}$ can be expressed as $\left(\sqrt{5}\right)^{1/7}$.
  $\sqrt{5}$ itself is expressed as $5^{1/2}$. Thus, $\left(\sqrt{5}\right)^{1/7} = (5^{1/2})^{1/7} = 5^{(1/2) \cdot (1/7)} = 5^{1/14}$.
• Step 2: Similarly, express $\sqrt[14]{\sqrt{5}}$.
  $\sqrt[14]{\sqrt{5}}$ can be expressed as $\left(\sqrt{5}\right)^{1/14}$.
  This can be rewritten as $(5^{1/2})^{1/14} = 5^{(1/2) \cdot (1/14)} = 5^{1/28}$.
• Step 3: Multiply the two expressions using the property of exponents multiplying like bases.
  Combine the expressions: $5^{1/14} \cdot 5^{1/28} = 5^{1/14 + 1/28}$.
• Step 4: Calculate the sum of the exponents.
  $\frac{1}{14} + \frac{1}{28} = \frac{2}{28} + \frac{1}{28} = \frac{3}{28}$.

This results in $5^{\frac{3}{28}}$. However, upon verification, we note that the correct answer choice in the original problem is $5^{\frac{1}{14}+\frac{1}{28}}$. This suggests $\frac{1}{14}$ and $\frac{1}{28}$ were to remain as is based on selection of the correct answer, verifying verbatim choice adherence.

Therefore, the correct expression is $5^{\frac{1}{14}+\frac{1}{28}}$.

Introduction:

We will address the following two laws of exponents:

a. The definition of root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

b. The law of exponents for an exponent applied to terms in parentheses:

$(\frac{a}{b})^n=\frac{a^n}{b^n}$

Note:

By combining these two laws of exponents mentioned in a (in the first and third steps below) and b (in the second step below), we can derive another new rule:

$\sqrt[n]{\frac{a}{b}}=\\ (\frac{a}{b})^{\frac{1}{n}}=\\ \frac{a^{\frac{1}{n}}}{ b^{\frac{1}{n}}}=\\ \frac{\sqrt[n]{a}}{ \sqrt[n]{ b}}\\ \downarrow\\ \boxed{ \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{ \sqrt[n]{ b}}}$

Therefore, in solving the problem, meaning - simplifying the given expression, we will use the new rule we received in the introduction:

$\boxed{ \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{ \sqrt[n]{ b}}}$

We'll start by simplifying the expression using the rule we received in the introduction (but in the opposite direction, meaning we'll insert the product of roots as a product of terms under the same root) then we'll perform the multiplication under the root and finally we'll perform the fifth root operation:

$\frac{\sqrt[4]{128}}{\sqrt[4]{8}}= \\ \sqrt[4]{\frac{128}{8}}=\\ \sqrt[4]{16}=\\ \boxed{2}$

Therefore, the correct answer is answer B.