Product Property of Square Roots: Using multiple rules

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 ) and b (in the second step ), we can obtain a new rule:

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

Specifically for the fourth root we obtain the following:

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

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

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

Specifically for the fourth root we obtain the following:

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

Therefore, in solving the problem, meaning - in simplifying the given expression, we will use the two new rules that we studied 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 that we studied in the introduction (1) (however this time in the opposite direction, meaning we'll insert the multiplication of roots as a multiplication of terms under the same root) Then we'll proceed to 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 to simplify the fraction, using the rule we received in the introduction (2) (however in the opposite direction, meaning we'll insert the division of roots as a division of terms under the same root) Then we'll proceed to 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:

$10$

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 ) and b' (in the second step ), we can obtain a new rule:

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

Specifically for the fourth root we obtain the following:

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

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

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

Specifically for the fourth root we obtain the following:

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

Therefore, in solving the problem, that is - in simplifying the given expression, we apply the two new rules that we studied 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 that we studied in the introduction (1) (however this time in the opposite direction, meaning we insert the multiplication of roots as a multiplication of terms under the same root) we then proceed to perform the multiplication under the root in the numerator:

$\frac{\sqrt{70}\cdot\sqrt{10}}{\sqrt{7}}= \\ \frac{\sqrt{70\cdot10}}{\sqrt{7}}= \\ \frac{\sqrt{700}}{\sqrt{7}}= \\$Continue to simplify the fraction, using the rule that we studied in the introduction (2) ( in the opposite direction, meaning we'll insert the division of roots as a division of terms under the same root) we'll then proceed to 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'.

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) and b (in the second step ), 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 apply the new rule studied 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 studied in the introduction (however this time in the opposite direction, meaning we'll insert the product of roots as a product of terms under the same root) We'll then proceed to 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.