Problem 40 - Precalculus Mathematics for Calculus

Question

Multiply the algebraic expressions using a Special Product Formula and simplify. $$(\sqrt{y}+\sqrt{2})(\sqrt{y}-\sqrt{2})$$

Step-by-Step Solution

Answer

The simplified expression is \(y - 2\).

Step 1: Identify the Special Product Formula

The expression \((\sqrt{y}+\sqrt{2})(\sqrt{y}-\sqrt{2})\) fits the form of the difference of squares formula, which is \((a + b)(a - b) = a^2 - b^2\). Here, \(a = \sqrt{y}\) and \(b = \sqrt{2}\).

Step 2: Apply the Formula

Using the difference of squares formula, substitute \(a = \sqrt{y}\) and \(b = \sqrt{2}\) to get \((\sqrt{y})^2 - (\sqrt{2})^2\).

Step 3: Simplify Each Term

First, calculate \((\sqrt{y})^2 = y\), since squaring a square root gives the original number. Then calculate \((\sqrt{2})^2 = 2\).

Step 4: Write the Simplified Expression

Subtract the second term from the first to get \(y - 2\). This is the simplified form of the original expression.

Key Concepts

Difference of Squares

The difference of squares is a specific pattern that comes in handy when multiplying algebraic expressions. It follows the formula

• \((a + b)(a - b) = a^2 - b^2\)

This formula is a quick way to simplify expressions without expanding everything completely. In our exercise, the expression \((\sqrt{y}+\sqrt{2})(\sqrt{y}-\sqrt{2})\) fits perfectly into this formula.
Here, \(a = \sqrt{y}\) and \(b = \sqrt{2}\). By recognizing this pattern, you can directly write down \(a^2 - b^2\).
This is an efficient technique that saves time and reduces the complexity of calculations. So, whenever you see a term like \((something + another)(something - another)\), try to spot if it's a difference of squares.

Radical Expressions

Radical expressions are expressions involving roots, like square roots or cube roots. In our exercise, we dealt with square roots: \(\sqrt{y}\) and \(\sqrt{2}\). Understanding radicals is crucial because they appear frequently in algebra.
When you square a square root, it simply cancels out the root operation, leaving the radicand (the number inside the root) by itself. For example:

• \((\sqrt{y})^2 = y\)
• \((\sqrt{2})^2 = 2\)

Working with radical expressions can seem tricky at first. Just remember the key rule: squaring a square root removes the radical. This is what simplifies the process and helps you handle these kinds of expressions effectively.

Simplifying Expressions

Simplifying expressions involves reducing an expression to its simplest form. In our original expression, after applying the difference of squares formula, we had two terms to simplify: \((\sqrt{y})^2\) and \((\sqrt{2})^2\).
After simplifying those terms, we were left with \(y - 2\), which is the simplest form of the expression. Simplification makes expressions easier to understand and work with, whether you're solving equations or performing further operations.
Here are some tips on simplifying:

• Look for patterns or formulas like the difference of squares.
• Always combine like terms, if possible.
• Simplify radicals by squaring them when necessary.

Keep practicing these steps to boost your confidence in handling both simple and complex expressions.