Solve for X: √2 × √3 = x/√6 Radical Equation

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:

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

Note:

By combining these two laws of exponents mentioned in a (in the first and third steps ahead) and b (in the second step ahead), 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}}$$$

$$Therefore, we will proceed with solving the problem as follows:

$\frac{x}{\sqrt{6}} = \sqrt{2}\cdot\sqrt{3}$$$

First, we'll eliminate the fraction line, which we'll do by multiplying both sides of the equation by the common denominator which is- $\sqrt{6}$:

$\frac{x}{\sqrt{6}} = \sqrt{2}\cdot\sqrt{3} \hspace{6pt}\text{/}\cdot\sqrt{6}\\ x=\sqrt{2}\cdot\sqrt{3}\cdot \sqrt{6}$

Let's continue and simplify the expression on the left side of the equation, using the rule we received in the introduction:

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

(which of course also applies to multiplication between numbers under a root), next we'll perform the multiplication under the root:

$x=\sqrt{2}\cdot\sqrt{3}\cdot\sqrt{6} \\ x=\sqrt{2\cdot3\cdot6} \\ x=\sqrt{36}\\ \boxed{x=6}$

In the final step, we used the known fourth root of the number 36,

Let's summarize the solution of the equation:

$\frac{x}{\sqrt{6}} = \sqrt{2}\cdot\sqrt{3} \hspace{6pt}\text{/}\cdot\sqrt{6}\\ x=\sqrt{2}\cdot\sqrt{3}\cdot \sqrt{6} \\ x=\sqrt{36}\\ \boxed{x=6}$

$$Therefore, the correct answer is answer a.