For \(x = \frac{{4\sqrt 6 }}{{\sqrt 2 + \sqrt 3 }},\)what is the value of \(\frac{{x + 2\sqrt 2 }}{{x - 2\sqrt 2 }} + \frac{{x\; + \;2\sqrt 3 }}{{x - 2\sqrt 3 }}?\)
2
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
| Step | Action | Result |
|---|---|---|
| 1 | Rewrite \(x\) and find \(\frac{x}{2\sqrt 2}\) | \(\frac{x}{2\sqrt 2} = \frac{2\sqrt 3}{\sqrt 2 + \sqrt 3}\) |
| 2 | Apply Componendo & Dividendo to \(\frac{x}{2\sqrt 2}\) | \(\frac{x + 2\sqrt 2}{x - 2\sqrt 2} = \frac{3\sqrt 3 + \sqrt 2}{\sqrt 3 - \sqrt 2}\) |
| 3 | Rewrite \(x\) and find \(\frac{x}{2\sqrt 3}\) | \(\frac{x}{2\sqrt 3} = \frac{2\sqrt 2}{\sqrt 2 + \sqrt 3}\) |
| 4 | Apply Componendo & Dividendo to \(\frac{x}{2\sqrt 3}\) | \(\frac{x + 2\sqrt 3}{x - 2\sqrt 3} = \frac{3\sqrt 2 + \sqrt 3}{\sqrt 2 - \sqrt 3}\) |
| 5 | Rewrite the second term | \(\frac{x + 2\sqrt 3}{x - 2\sqrt 3} = - \frac{3\sqrt 2 + \sqrt 3}{\sqrt 3 - \sqrt 2}\) |
| 6 | Add the simplified terms | \(\frac{3\sqrt 3 + \sqrt 2}{\sqrt 3 - \sqrt 2} - \frac{3\sqrt 2 + \sqrt 3}{\sqrt 3 - \sqrt 2} = \frac{2\sqrt 3 - 2\sqrt 2}{\sqrt 3 - \sqrt 2}\) |
| 7 | Simplify the final fraction | \(\frac{2(\sqrt 3 - \sqrt 2)}{\sqrt 3 - \sqrt 2} = 2\) |
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.
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