Problem 92 - College Algebra with Modeling and Visualization

Question

Multiply and simplify. $$ (\sqrt{a b}-3)(\sqrt{a b}+3) $$

Step-by-Step Solution

Answer

The simplified expression is \(a b - 9\).

Step 1: Recognize the Expression Type

Observe that the expression \((\sqrt{a b}-3)(\sqrt{a b}+3)\) is in the form \((x-y)(x+y)\), which is a difference of squares formula.

Step 2: Apply the Difference of Squares Formula

The difference of squares formula states that \((x-y)(x+y) = x^2 - y^2\). Let \(x = \sqrt{a b}\) and \(y = 3\). Substitute these into the formula to obtain \((\sqrt{a b})^2 - 3^2\).

Step 3: Simplify the Squares

Calculate \((\sqrt{a b})^2\) which is equal to \(a b\), and \(3^2\) which equals \(9\). Substitute these values back into the expression to get \(a b - 9\).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They are fundamental in algebra and are used to represent quantities and relationships. For instance, in the expression \(\sqrt{ab} - 3\), the square root of the product of \(a\) and \(b\) is directly followed by a subtraction, where 3 is subtracted from it.

Understanding the structure and components of algebraic expressions is key to performing operations such as addition, subtraction, multiplication, and division.

• Variables: Symbols like \(a\) and \(b\) represent quantities that can change.
• Constants: Numbers like \(3\) are fixed values in the expression.
• Radical Expression: \(\sqrt{ab}\) represents a radical or root form.

Breaking down algebraic expressions into parts helps simplify them and allows for more manageable manipulation for solving equations or inequalities.

Simplifying Radicals

Simplifying radicals involves reducing the expression under a root to its simplest form. This process allows for easier manipulation and operation when the expression involves roots.

In our exercise, \(\sqrt{ab}\) represents a radical. The objective is to either make the expression less complex or integrate it seamlessly with other algebraic components.

• Recognize Perfect Squares: If \(a\) and \(b\) are perfect squares, \(\sqrt{ab}\) can be simplified directly to its numerical form.
• Factor and Simplify: If potential factors exist within \(ab\), break them down to simplify \(\sqrt{ab}\) further.
• Perform Operations: Combine like radicals and simplify using arithmetic operations.

Ensure your radicals are in the simplest form for precise calculations in broader algebraic expressions.

Polynomial Operations

Polynomial operations involve performing addition, subtraction, multiplication, and division on polynomial expressions. These operations are essential for solving equations and simplifying larger expressions.

When handling polynomial operations, such as multiplication in this exercise, recognizing patterns like the difference of squares formula can simplify computation drastically.

• Difference of Squares: This is a special product pattern \((x-y)(x+y) = x^2 - y^2\).
• Simplification: Multiply and simplify terms like \(x^2\) and \(y^2\) independently.
• Reorganization: Combine terms after multiplication to form a single expression or polynomial.

In our exercise, after identifying the difference of squares pattern, we simplified the expression to a single polynomial, \(ab - 9\). Understanding these operations paves the way for solving more intricate polynomial problems efficiently.