Problem 32 - 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. $$ (7+\sqrt{5 y})(3-\sqrt{5 y}) $$

Step-by-Step Solution

Answer

The simplest form is \(49 - 5y\).

Step 1: Identify the Expression Form

The expression given is \((7+\sqrt{5 y})(3-\sqrt{5 y})\). This is a product that can be simplified using the difference of squares formula since it matches the form of \((a+b)(a-b) = a^2 - b^2\).

Step 2: Assign Terms to Variables

For the formula \((a+b)(a-b) = a^2 - b^2\), identify a = 7 and b = \sqrt{5y}\, corresponding to \((7+\sqrt{5y})(3-\sqrt{5y})\).

Step 3: Apply the Difference of Squares Formula

Substitute the values into the difference of squares formula: \(a^2 - b^2\). Here, \(a^2 = 7^2 = 49\) and \(b^2 = (\sqrt{5y})^2 = 5y\).

Step 4: Simplify the Expression

Substitute the squared terms into the expression: \(49 - 5y\). The simplified form of the given expression \((7+\sqrt{5y})(3-\sqrt{5y})\) is thus \(49 - 5y\).

Key Concepts

Rational Expression

A rational expression is a fraction where the numerator and the denominator are both polynomials. In our example, \( (7+\sqrt{5y})(3-\sqrt{5y}) \), we engage in simplifying a product that appears in a fractional structure when expressed over a shared denominator.
However, unlike a typical rational expression, the given problem involves a radicand, leading us to explore its simplification without explicitly forming a rational expression.
By understanding the underlying concepts of rational expressions and their simplification, we can appreciate the process and solutions involved, as these techniques help significantly in simplifying complex-looking problems.

Simplification

Simplification is about making expressions as manageable as possible. In this problem, the given expression is in the form of \((a+b)(a-b)\), which is perfect for the difference of squares formula. This formula states that \((a+b)(a-b) = a^2 - b^2\).
The goal is to transform an expression into its simplest form. In our example:- We identify \(a\) as 7 and \(b\) as \(\sqrt{5y}\).- Then, we use the difference of squares to quickly simplify. Squaring both terms separately, we get \(a^2 = 49\) and \(b^2 = 5y\).- Substituting back into our formula gives \(49 - 5y\).
Simplification helps to reduce computational complexity, making it easier to handle advanced mathematical operations.

Radicand Expression

A radicand expression involves terms under a square root or any root symbol, like \(\sqrt{5y}\) in this problem. The radicand is the expression inside the radical.
In our exercise:- \(\sqrt{5y}\) represents the radicand expression.- The problem specifies variables with an even index, which in this case, are non-negative. This means \(5y\) is assumed to have a suitable domain to ensure no negative results under the square root, ensuring real numbers.- Understanding radicand expressions helps in recognizing how they appear in simplification processes, particularity when using formulas like the difference of squares.
Handling these expressions involves careful application of algebraic rules, ensuring each component fits naturally with formulas used in simplification.