Animated Solution for Mathematics - Basic Mathematics: The expression $\frac{12}{3+\sqrt{5}+2\sqrt{2}}$ is equal to

Welcome, future engineer. Today, we are going to demystify a problem that often intimidates students at first glance: the rationalization of a complex denominator. We are looking at the expression:

At first, it looks like a mess of roots, but I want you to see it as a puzzle waiting to be solved. The key to mastering JEE Advanced problems is not just knowing the formulas, but understanding the strategy behind them.

When you see three terms in a denominator, your instinct might be to panic. But remember, algebra is about structure. We can treat this as a binomial problem by grouping.

Let us define $a=3+\sqrt{5}$ and $b=2\sqrt{2}$. Now, our denominator is simply $a+b$.

This is a classic JEE trick: reducing a complex expression into a familiar form. By grouping, we have turned a three-term nightmare into a two-term problem that we know how to handle.

Now that we have our binomial $a+b$, we know exactly what to do. We multiply the numerator and the denominator by the conjugate, $a-b$.

So, we multiply by $(3+\sqrt{5})-2\sqrt{2}$. This step is crucial because it allows us to use the identity $(a+b)(a-b)=a^2-b^2$.

When we apply this to the denominator, we get:

Let us expand this carefully. $(3+\sqrt{5}{)}^2$ becomes $9+5+6\sqrt{5}$, which is $14+6\sqrt{5}$. Subtracting $(2\sqrt{2}{)}^2$, which is $8$, we get:

Factoring out the $6$, we are left with $6(1+\sqrt{5})$.

We have made significant progress, but we are not at the finish line yet. Our expression is now:

Simplifying the $12$ and $6$, we get:

The denominator is still irrational! This is where many students give up, but you are not one of them. We simply perform the rationalization process again by multiplying the numerator and denominator by the conjugate of $1+\sqrt{5}$, which is $\sqrt{5}-1$.

Now, we multiply the numerator $2(3+\sqrt{5}-2\sqrt{2})$ by $(\sqrt{5}-1)$. Expanding this, we get:

Combining like terms, this simplifies to $2(2+2\sqrt{5}+2\sqrt{2}-2\sqrt{10})$. Factoring out the $2$, we get:

The denominator, $(\sqrt{5}+1)(\sqrt{5}-1)$, becomes $5-1=4$. The $4$ in the numerator and the $4$ in the denominator cancel out perfectly, leaving us with the beautiful final answer:

This is the power of systematic thinking. You didn't just solve a problem; you navigated a complex landscape with precision. Keep this mindset, and no problem will ever be too difficult for you.