Problem 83 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Recognize the Expression Pattern

The given problem is \[ (\(\sqrt{a} - b\))(\(\sqrt{a} + b\))\], which is of the form of the difference of squares pattern \((x-y)(x+y)\). The difference of squares formula says \((x-y)(x+y)= x^2 - y^2\).

Step 2: Identify Terms for Difference of Squares

Identify \(x = \sqrt{a}\) and \(y = b\). So, the expression can be simplified using the difference of squares formula: \((\sqrt{a})^2 - b^2\).

Step 3: Apply the Formula

Substitute \(x\) and \(y\) into the formula: \((\sqrt{a})^2 - b^2 = a - b^2\).

Step 4: Simplified Result

Simplify the expression \(a - b^2\). This is the final, simplest form of the expression. No further calculations are needed.

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential part of algebra that helps make complex or lengthy mathematical expressions easier to work with. In algebra, simplification involves rewriting an expression in a more concise or readable form without changing its value. This process often makes calculations more straightforward and reduces the possibility of errors.
When simplifying expressions involving the difference of squares, you use specific algebraic identities which simplify computations and highlight the underlying structure. For example, in the original exercise, we encountered an expression

• (\(\sqrt{a} - b\))(\(\sqrt{a} + b\))

This expression matches the standard identity for the difference of squares:

• (\(x - y\))(\(x + y\)) = \(x^2 - y^2\).

Using these identities not only simplifies the expression but also makes it recognizable, allowing for quick solutions to complex problems.

Algebraic Operations

Algebraic operations are the basic building blocks in mathematics that allow us to manipulate and solve equations or expressions. These operations include addition, subtraction, multiplication, and division, as well as the application of various algebraic formulas and identities.
In the problem we're discussing, we employ a well-known algebraic identity, the difference of squares, which is particularly useful when simplifying expressions like

• (\(\sqrt{a} - b\))(\(\sqrt{a} + b\)).

The use of this identity allows us to transform and simplify the expression by recognizing it as a product of two conjugate binomials, leading to

• a more straightforward expression \(a - b^2\).

Understanding these operations within their various contexts helps in navigating more complex algebra problems later on.

Problem-Solving

Problem-solving in algebra often involves recognizing patterns and applying the correct algebraic rules and identities to find a solution. The difference of squares is a frequent pattern that, once recognized, allows for effective and efficient problem-solving.
Approaching a problem like

• (\(\sqrt{a} - b\))(\(\sqrt{a} + b\))

requires identifying it as a difference of squares situation. By realizing

• x = \(\sqrt{a}\) and y = \(b\),

we can directly apply our knowledge to find

• a simple solution: \(a - b^2\).

Ultimately, problem-solving in algebra is about recognizing these kinds of patterns, understanding the rules that apply, and correctly applying them to simplify and solve expressions or equations.