91Ó°ÊÓ

The distributive property is a fundamental algebraic rule used to simplify expressions. It states that for any numbers or variables, \(a, b,\) and \(c\), the expression \(a(b + c)\) is equal to \(ab + ac\). This property ensures that each term inside a parenthesis is multiplied by the term outside.
In our exercise, we have two binomials, \((x + \sqrt{2})(x + \sqrt{3})\).
To simplify this, we distribute each term in the first binomial\((x + \sqrt{2})\) to every term in the second binomial\((x + \sqrt{3})\). This step is crucial for expanding the product of these binomials into a form where we can further simplify.

Multiplying binomials involves expanding them by applying the distributive property step by step. For our given problem \((x + \sqrt{2})(x + \sqrt{3})\), we start by distributing each term inside the first parenthesis to every term in the second parenthesis:

1. Multiply \(x\) in the first binomial with each term in the second binomial:
\ x(x + \sqrt{3}) = x^2 + x\sqrt{3} \
2. Now, multiply \(\sqrt{2}\) in the first binomial with each term in the second binomial:
\ \sqrt{2}(x + \sqrt{3}) = x \sqrt{2} + \sqrt{2} \cdot \sqrt{3} \
3. Expand the above multiplications: \ ( x^2 + x\sqrt{3} + x\sqrt{2} + \sqrt{6} ) \ Here we note that multiplying square roots results in another square root, such as \(\sqrt{2} \cdot \sqrt{3} = \sqrt{6}\).

After expanding the binomials, we need to combine like terms to simplify the final expression.
Like terms are terms that have the same variable raised to the same power. In our simplified expression \( x^2 + x\sqrt{3} + x\sqrt{2} + \sqrt{6} \), we observe that:

• \( x^2 \) is a standalone term.
• \( x\sqrt{3} \) and \( x\sqrt{2} \) are unlike terms due to the different radicals but can be grouped as terms containing x.
• \( \sqrt{6} \) is also a standalone term as it has a different square root.
  Consequently, we find that none of the terms above can be combined further. This will be the simplified version of the expanded expression.