Problem 33 - Algebra 2 and Trigonometry

Question

In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ (7+\sqrt{5 b})(7-\sqrt{5 b}) $$

Step-by-Step Solution

Answer

The simplest form of the expression is \(49 - 5b\).

Step 1: Identify the form of the expression

The given expression is \((7+\sqrt{5b})(7-\sqrt{5b})\), which follows the \(a^2-b^2\) pattern, known as a difference of squares. Here, \(a = 7\) and \(b = \sqrt{5b}\).

Step 2: Apply the Difference of Squares Formula

The difference of squares formula is \(a^2 - b^2\), where \(a = 7\) and \(b = \sqrt{5 b}\). Substitute these values into the formula to find:\[7^2 - (\sqrt{5b})^2\].

Step 3: Calculate the Squares

Evaluate each part of the expression:\[7^2 = 49\].The square of a square root cancels out the square root, so:\[(\sqrt{5b})^2 = 5b\].

Step 4: Subtract the Results

Now, perform the subtraction as per the formula:\[49 - 5b\].This results in the simplified form of the expression.

Key Concepts

Radicals

Radicals, often represented by the square root symbol (\(\sqrt{}\)), indicate a value that, when multiplied by itself a specific number of times, gives the original number under the radical. Radicals can involve numbers or algebraic expressions. For example, \(\sqrt{5b}\) involves both a numerical value and a variable. The process of dealing with radicals often includes simplifying them, similar to simplifying fractions.

When dealing with radicals, keep the following in mind:

• The square root of a product, such as \(\sqrt{16}\), results in the number that, when squared, equals the product, here \(4\).
• The square root of a variable, such as \(\sqrt{x^2}\), equals the variable \(x\) when \(x\) is non-negative.
• Radicals appear frequently in operations like addition, subtraction, multiplication, and division, often requiring extra steps for simplification.

Understanding how to manipulate radicals is essential for solving algebraic problems efficiently.

Simplification

Simplification is a core skill in algebra that involves reducing expressions to their simplest form without changing their value. This is crucial in making algebraic expressions easier to understand and work with. Simplifying generally includes reducing coefficients, canceling common factors, and unifying like terms.

When applied to

• Numbers: Factorize both numbers involved to find the greatest common divisor (GCD) and divide each by this GCD.
• Expressions: Use identity formulas such as the difference of squares, or combine like terms, to streamline expressions.
• Fractions and Radicals: Break down numerators and denominators to lowest terms and simplify radicals by extracting perfect squares.

Each step of simplification makes the expression easier to handle, which is especially useful in solving equations or comparing expressions.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operators (like addition, subtraction, etc.) combined in a meaningful way. They form the foundation of algebra and are used to represent relationships between quantities.

Key components include:

• Constants: Fixed numerical values (like \(7\) in our case).
• Variables: Symbols that stand in for unknown values (\(b\) is such a variable here).
• Operators: Signs like +, -, *, and / that dictate the operations to perform.

In the given exercise, the expression is \((7+\sqrt{5b})(7-\sqrt{5b})\). Here, it uses the difference of squares identity to simplify, indicating the crucial interplay of constants and variables in algebraic expressions. Mastery of these expressions is essential for success in algebra and helps you transition seamlessly into more advanced mathematics.