Problem 36 - Algebra 1: Concepts and Skills

Question

Simplify the expression using the sum and difference pattern. $$ (\sqrt{2}+\sqrt{3})(\sqrt{2}-\sqrt{3}) $$

Step-by-Step Solution

Answer

The simplified form of the expression \((\sqrt{2}+\sqrt{3})(\sqrt{2}-\sqrt{3})\) is -1.

Step 1: Identify a and b

In this exercise, we can identify a as \(\sqrt{2}\) and b as \(\sqrt{3}\). This is based off the format of the problem, as it has two terms, one positive and the other negative, being multiplied together.

Step 2: Apply the sum and difference formula

We apply the sum and difference formula, substituting a and b with \(\sqrt{2}\) and \(\sqrt{3}\) into the formula \((a + b)(a - b) = a^2 - b^2\). So the expression becomes \((\sqrt{2})^2 - (\sqrt{3})^2\).

Step 3: Simplify the resulting expression

Simplify the expression. We square the square roots to get rid of them, which gives us 2 - 3.

Step 4: Find the final result

The final result comes out to be -1, since 2 - 3 equals -1.

Key Concepts

Simplifying Expressions

Simplifying expressions often involves recognizing patterns or structures that allow us to reduce complexity. In this problem, given the expression \[ (\sqrt{2}+\sqrt{3})(\sqrt{2}-\sqrt{3}) \], the idea is to apply a known algebraic identity.One effective pattern is the sum and difference rule, which can simplify expressions quickly. This identity states that for any two real numbers, the product of their sum and difference results in the difference of their squares:\[ (a + b)(a - b) = a^2 - b^2 \].Using this formula minimizes the effort to expand multiple brackets individually and notably, it converts our expression into a simpler one with no parentheses. This pattern not only speeds up the process of solving problems but also reduces the chance of arithmetic mistakes.

Difference of Squares

The difference of squares is a powerful concept in algebra used to simplify expressions and equations. It takes the form:\[ a^2 - b^2 \], where simply recognizing this pattern allows for direct simplification to \[ (a + b)(a - b) \], or conversely, using the identity, to simplify a product.In our exercise, the expression \[ (\sqrt{2} + \sqrt{3})(\sqrt{2} - \sqrt{3}) \] was identified as fitting this pattern. Here, \[ a = \sqrt{2} \] and \[ b = \sqrt{3} \]. Applying the formula to simplify, we directly obtain:\[ (\sqrt{2})^2 - (\sqrt{3})^2. \]This illustrates how the difference of squares isn't just about expanding and factoring, but also precisely recognizing when to apply it, for simplification.

Square Roots

Square roots denote the value that, when multiplied by itself, yields the original number. For example, the square root of 4 is 2, because \[ 2\times2 = 4 \]. In our given problem, we deal with \[ \sqrt{2} \] and \[ \sqrt{3} \].The beauty of square roots arises when they are squared. Squaring a square root cancels the square and the square root, returning us to the original number. Thus, \[ (\sqrt{2})^2 = 2 \] and \[ (\sqrt{3})^2 = 3 \].In the exercise at hand, squaring the square roots within each term straightforwardly simplifies them to their base values, enabling easier arithmetic. This step is intuitive if you keep in mind the inverse relationship between squaring and taking a square root, which simplifies expressions elegantly.