To prove that $\sqrt{3} + \sqrt{5}$ is irrational, we will use a proof by contradiction. We assume the opposite of what we want to prove, show that this assumption leads to a contradiction, and therefore conclude that our original statement must be true.

Step 1: Assume the opposite.
Let us assume that $\sqrt{3} + \sqrt{5}$ is a rational number. By definition, a rational number can be expressed as a fraction $\frac{a}{b}$ , where $a$ and $b$ are integers and $b \neq 0$ .

So, let $\sqrt{3} + \sqrt{5} = \frac{p}{q}$ , where $p$ and $q$ are co-prime integers (i.e., their HCF is 1) and $q \neq 0$ .

Step 2: Manipulate the equation.
We can rearrange the equation to isolate one of the square roots. $\sqrt{3} = \frac{p}{q} - \sqrt{5}$

Step 3: Square both sides.
Squaring both sides will help us eliminate the square roots. ${(\sqrt{3})}^{2} = {(\frac{p}{q} - \sqrt{5})}^{2}$
This simplifies to: $3 = {(\frac{p}{q})}^{2} - 2 \times \frac{p}{q} \times \sqrt{5} + {(\sqrt{5})}^{2}$
$3 = \frac{p^{2}}{q^{2}} - \frac{2 p}{q} \sqrt{5} + 5$

Step 4: Isolate the remaining square root term.
Let's get the term containing $\sqrt{5}$ by itself. $\frac{2 p}{q} \sqrt{5} = \frac{p^{2}}{q^{2}} + 5 - 3$
$\frac{2 p}{q} \sqrt{5} = \frac{p^{2}}{q^{2}} + 2$
Now, solve for $\sqrt{5}$ : $\sqrt{5} = \frac{q}{2 p} (\frac{p^{2}}{q^{2}} + 2)$
$\sqrt{5} = \frac{q}{2 p} \times \frac{p^{2}}{q^{2}} + \frac{q}{2 p} \times 2$
$\sqrt{5} = \frac{p}{2 q} + \frac{q}{p}$

Step 5: Identify the contradiction.
The right-hand side of the equation, $\frac{p}{2 q} + \frac{q}{p}$ , is a rational number because it is the sum of two rational numbers (since p and q are integers). However, the left-hand side, $\sqrt{5}$ , is a well-known irrational number.

A rational number cannot be equal to an irrational number. This is a contradiction.

Conclusion:
Since our initial assumption that $\sqrt{3} + \sqrt{5}$ is rational has led to a contradiction, the assumption must be false. Therefore, $\sqrt{3} + \sqrt{5}$ is an irrational number.

Related Concepts & Formulae

Rational Numbers: A number that can be expressed in the form p/q, where p and q are integers and q ≠ 0. Their decimal expansions are either terminating or repeating.

Irrational Numbers: A number that cannot be expressed as a simple fraction. Its decimal expansion is non-terminating and non-repeating. Examples include √2, √3, √5, and π.

Proof by Contradiction: A powerful mathematical technique where you assume the opposite of what you want to prove. If this assumption leads to an impossible situation (a contradiction), then your original statement must be true.

Key Property Used: The sum of a rational number and an irrational number is irrational. The product of a non-zero rational number and an irrational number is irrational.