Evaluating Algebraic Expressions with Radicals

We are given the value of $x$ as $x = \frac{{4\sqrt 6 }}{{\sqrt 2 + \sqrt 3 }}$ and asked to find the value of the expression $\frac{{x + 2\sqrt 2 }}{{x - 2\sqrt 2 }} + \frac{{x\; + \;2\sqrt 3 }}{{x - 2\sqrt 3 }}$.

Let's analyze the given value of $x$. We can rewrite $4\sqrt 6$ as $4 \times \sqrt{2 \times 3} = 4\sqrt 2 \sqrt 3$. So, $x = \frac{4\sqrt 2 \sqrt 3}{\sqrt 2 + \sqrt 3}$.

The expression we need to evaluate has terms in the form $\frac{a+b}{a-b}$. This structure is strongly related to the componendo and dividendo rule.

Recall the componendo and dividendo rule: If $\frac{a}{b} = \frac{c}{d}$, then $\frac{a+b}{a-b} = \frac{c+d}{c-d}$. We can also use this rule in reverse or apply it to rearranged forms of the initial ratio.

Step 1: Evaluate the First Term $\frac{{x + 2\sqrt 2 }}{{x - 2\sqrt 2 }}$

Let's consider the first term $\frac{{x + 2\sqrt 2 }}{{x - 2\sqrt 2 }}$. This term suggests we should look at the ratio $\frac{x}{2\sqrt 2}$.

Using the given value of $x$:

$\frac{x}{2\sqrt 2} = \frac{\frac{4\sqrt 2 \sqrt 3}{\sqrt 2 + \sqrt 3}}{2\sqrt 2}$

$\frac{x}{2\sqrt 2} = \frac{4\sqrt 2 \sqrt 3}{2\sqrt 2 (\sqrt 2 + \sqrt 3)}$

$\frac{x}{2\sqrt 2} = \frac{2\sqrt 3}{\sqrt 2 + \sqrt 3}$

Now, applying the componendo and dividendo rule to $\frac{x}{2\sqrt 2} = \frac{2\sqrt 3}{\sqrt 2 + \sqrt 3}$:

$\frac{x + 2\sqrt 2}{x - 2\sqrt 2} = \frac{2\sqrt 3 + (\sqrt 2 + \sqrt 3)}{2\sqrt 3 - (\sqrt 2 + \sqrt 3)}$

Simplify the numerator and the denominator on the right side:

Numerator: $2\sqrt 3 + \sqrt 2 + \sqrt 3 = (2\sqrt 3 + \sqrt 3) + \sqrt 2 = 3\sqrt 3 + \sqrt 2$

Denominator: $2\sqrt 3 - \sqrt 2 - \sqrt 3 = (2\sqrt 3 - \sqrt 3) - \sqrt 2 = \sqrt 3 - \sqrt 2$

So, the first term is:

$\frac{x + 2\sqrt 2}{x - 2\sqrt 2} = \frac{3\sqrt 3 + \sqrt 2}{\sqrt 3 - \sqrt 2}$

Step 2: Evaluate the Second Term $\frac{{x\; + \;2\sqrt 3 }}{{x - 2\sqrt 3 }}$

Next, let's consider the second term $\frac{{x\; + \;2\sqrt 3 }}{{x - 2\sqrt 3 }}$. This term suggests looking at the ratio $\frac{x}{2\sqrt 3}$.

Using the given value of $x$:

$\frac{x}{2\sqrt 3} = \frac{\frac{4\sqrt 2 \sqrt 3}{\sqrt 2 + \sqrt 3}}{2\sqrt 3}$

$\frac{x}{2\sqrt 3} = \frac{4\sqrt 2 \sqrt 3}{2\sqrt 3 (\sqrt 2 + \sqrt 3)}$

$\frac{x}{2\sqrt 3} = \frac{2\sqrt 2}{\sqrt 2 + \sqrt 3}$

Now, applying the componendo and dividendo rule to $\frac{x}{2\sqrt 3} = \frac{2\sqrt 2}{\sqrt 2 + \sqrt 3}$:

$\frac{x + 2\sqrt 3}{x - 2\sqrt 3} = \frac{2\sqrt 2 + (\sqrt 2 + \sqrt 3)}{2\sqrt 2 - (\sqrt 2 + \sqrt 3)}$

Simplify the numerator and the denominator on the right side:

Numerator: $2\sqrt 2 + \sqrt 2 + \sqrt 3 = (2\sqrt 2 + \sqrt 2) + \sqrt 3 = 3\sqrt 2 + \sqrt 3$

Denominator: $2\sqrt 2 - \sqrt 2 - \sqrt 3 = (2\sqrt 2 - \sqrt 2) - \sqrt 3 = \sqrt 2 - \sqrt 3$

So, the second term is:

$\frac{x + 2\sqrt 3}{x - 2\sqrt 3} = \frac{3\sqrt 2 + \sqrt 3}{\sqrt 2 - \sqrt 3}$

Step 3: Add the Two Terms

We need to find the sum of the two terms:

Expression = $\frac{x + 2\sqrt 2}{x - 2\sqrt 2} + \frac{x + 2\sqrt 3}{x - 2\sqrt 3}$

Substitute the simplified forms from Step 1 and Step 2:

Expression = $\frac{3\sqrt 3 + \sqrt 2}{\sqrt 3 - \sqrt 2} + \frac{3\sqrt 2 + \sqrt 3}{\sqrt 2 - \sqrt 3}$

Notice that the denominators $\sqrt 3 - \sqrt 2$ and $\sqrt 2 - \sqrt 3$ are negatives of each other. We can write $\sqrt 2 - \sqrt 3 = -(\sqrt 3 - \sqrt 2)$.

So the second term can be rewritten as:

$\frac{3\sqrt 2 + \sqrt 3}{\sqrt 2 - \sqrt 3} = \frac{3\sqrt 2 + \sqrt 3}{-(\sqrt 3 - \sqrt 2)} = - \frac{3\sqrt 2 + \sqrt 3}{\sqrt 3 - \sqrt 2}$

Now substitute this back into the sum:

Expression = $\frac{3\sqrt 3 + \sqrt 2}{\sqrt 3 - \sqrt 2} - \frac{3\sqrt 2 + \sqrt 3}{\sqrt 3 - \sqrt 2}$

Since the denominators are the same, we can combine the numerators:

Expression = $\frac{(3\sqrt 3 + \sqrt 2) - (3\sqrt 2 + \sqrt 3)}{\sqrt 3 - \sqrt 2}$

Carefully remove the parentheses in the numerator:

Expression = $\frac{3\sqrt 3 + \sqrt 2 - 3\sqrt 2 - \sqrt 3}{\sqrt 3 - \sqrt 2}$

Group like terms in the numerator:

Expression = $\frac{(3\sqrt 3 - \sqrt 3) + (\sqrt 2 - 3\sqrt 2)}{\sqrt 3 - \sqrt 2}$

Simplify the terms in the numerator:

Expression = $\frac{2\sqrt 3 - 2\sqrt 2}{\sqrt 3 - \sqrt 2}$

Factor out the common factor 2 from the numerator:

Expression = $\frac{2(\sqrt 3 - \sqrt 2)}{\sqrt 3 - \sqrt 2}$

Cancel out the common factor $(\sqrt 3 - \sqrt 2)$ in the numerator and denominator:

Expression = $2$

Thus, the value of the expression is 2.

Revision Table: Key Steps Summary

Additional Information: Understanding Componendo and Dividendo

The componendo and dividendo rule is a useful tool when dealing with ratios and proportions. It states that if four quantities are in proportion, i.e., if $\frac{a}{b} = \frac{c}{d}$ (where $b \neq 0$ and $d \neq 0$), then they are also in proportion under componendo ($\frac{a+b}{b} = \frac{c+d}{d}$), dividendo ($\frac{a-b}{b} = \frac{c-d}{d}$), and componendo and dividendo combined ($\frac{a+b}{a-b} = \frac{c+d}{c-d}$, provided $a \neq b$ and $c \neq d$).

In this problem, we used the combined componendo and dividendo rule. By rearranging the given expression for $x$ to form a ratio involving $x$ and the terms in the denominators ($2\sqrt 2$ and $2\sqrt 3$), we were able to directly simplify the two parts of the expression we needed to evaluate. This method avoided needing to substitute the full complex expression for $x$ directly into the numerator and denominator of the target expression, which would have been much more complicated.