Problem 55 - Precalculus Mathematics for Calculus

Question

Perform the indicated operations and simplify. $$(\sqrt{a}-b)(\sqrt{a}+b)$$

Step-by-Step Solution

Answer

The simplified expression is \(a - b^2\).

Step 1: Recognize the Expression

The expression \((\sqrt{a} - b)(\sqrt{a} + b)\) is a product of conjugates.

Step 2: Recall the Conjugate Product Formula

When two conjugate expressions are multiplied, they follow the identity: \((x - y)(x + y) = x^2 - y^2\).

Step 3: Apply the Conjugate Product Formula

Apply the formula to the problem: \((\sqrt{a} - b)(\sqrt{a} + b) = (\sqrt{a})^2 - b^2\).

Step 4: Simplify the Expression

Simplify \((\sqrt{a})^2\) to \(a\), so the expression becomes \(a - b^2\).

Key Concepts

Square Roots

Understanding square roots is crucial when tackling problems like this one. A square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3, because 3 times 3 equals 9. They are often represented using a radical symbol:

• \( \sqrt{9} = 3 \)
• \( \sqrt{a} \) represents a number which when squared gives \( a \)

When dealing with algebraic expressions that include square roots, it's important to apply mathematical identities correctly to simplify the expression without making any assumptions about the values of the variables involved. In expressions like \( (\sqrt{a} - b)(\sqrt{a} + b) \), understanding how the square root affects multiplication and conjugation is key to simplifying it effectively. Always remember that (\( \sqrt{a})^2 = a \).
We use these properties to help us breakdown expressions involving roots, as in the next concept.

Simplifying Expressions

Simplifying expressions involves a systematic approach to make an expression easier to work with. The goal is to rewrite an expression in a simpler form that retains the same value. With an expression like \((\sqrt{a} - b)(\sqrt{a} + b)\), simplifying means applying known algebraic rules or identities to reduce the expression.

The expression, \((\sqrt{a} - b)(\sqrt{a} + b)\), can be quite complex at first glance. But recognizing this as a special form allows us to utilize the identity for multiplying conjugates, which greatly simplifies the calculation. After observing and identifying the expression as a product of conjugates, it guides us to apply the identity \((x - y)(x + y) = x^2 - y^2\).
Through this simplification process:

• First, identify which terms need squaring, such as \((\sqrt{a})^2 = a\).
• Recognize that \(b^2\) is simply \(b \times b\).
• Apply it into the formula \(x^2 - y^2\) for quick simplification.

Finally, \[ \text{Expression becomes } a - b^2.\]After these steps, the expression is simpler and easy to understand.

Algebraic Identities

In algebra, it's common to use fixed patterns known as identities to ease the manipulation of expressions. Algebraic identities are essential in simplifying complex algebraic problems and calculations. They represent a form of equality that holds true for all values of the involved variables.
A vital identity is the difference of squares:

• \((x - y)(x + y) = x^2 - y^2\)

This identity is especially helpful when dealing with conjugate pairs like \((\sqrt{a}-b)\) and \((\sqrt{a}+b)\). By recognizing an expression as the product of conjugates, we can directly use this identity to simplify:

• For our specific example, apply it to get \((\sqrt{a})^2 - b^2 = a - b^2\).

Each algebraic identity serves as a shortcut, and mastering them saves time and reduces errors. Whenever you encounter an expression, see if you can spot an identity to transform it into a more manageable form. This makes solving problems not only easier but also more interesting!