Solve for X: Square Root Fraction Equation √64/√4 = 2x

Introduction:

We will address the following two laws of exponents:

a. 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 the two laws of exponents mentioned in a (in the first and third stages below) and b (in the second stage below), we can derive another new rule:

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

Specifically for the fourth root we obtain the following:

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

Therefore, we can proceed to solve the problem:

$\frac{\sqrt{64}}{\sqrt{4}}=2x$

Let's start by simplifying the expression on the left side, using the new rule that we studied in the introduction:

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

( However this time 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:

$\frac{\sqrt{64}}{\sqrt{4}}=2x \\ \sqrt{\frac{64}{4}}=2x \\ \sqrt{16}=2x \\ 4=2x \\$In the final stage, we used the known fourth root of the number 16,

After simplifying the expression on the left side, to isolate the unknown, we'll divide both sides of the equation by its coefficient:

$4=2x\hspace{6pt}\text{/}:2 \\ 2=x \\ \downarrow\\ \boxed{x=2}$

Let's summarize the solution of the equation:

$\frac{\sqrt{64}}{\sqrt{4}}=2x \\ \sqrt{16}=2x \\ 4=2x \\ \downarrow\\ \boxed{x=2}$$$

Therefore, the correct answer is answer b.