Problem 40 - Algebra and Trigonometry

Question

Perform the indicated operations and simplify. $$ \left(x^{1 / 2}+y^{1 / 2}\right)\left(x^{1 / 2}-y^{1 / 2}\right) $$

Step-by-Step Solution

Answer

The simplified form is \(x - y\).

Step 1: Recognize the Formula

The expression \((x^{1/2} + y^{1/2})(x^{1/2} - y^{1/2})\) resembles the form of a difference of squares formula, \((a+b)(a-b) = a^2 - b^2\).

Step 2: Identify Parts of the Formula

Identify \(a\) and \(b\) in the expression where \(a = x^{1/2}\) and \(b = y^{1/2}\).

Step 3: Apply the Difference of Squares Formula

Using \((a+b)(a-b) = a^2 - b^2\), substitute \(a = x^{1/2}\) and \(b = y^{1/2}\): \[(x^{1/2} + y^{1/2})(x^{1/2} - y^{1/2}) = (x^{1/2})^2 - (y^{1/2})^2\].

Step 4: Simplify the Expressions

Calculate \((x^{1/2})^2 = x\) and \((y^{1/2})^2 = y\).

Step 5: Write the Simplified Result

The simplified form after applying the operations is \(x - y\).

Key Concepts

Algebraic Simplification

Algebraic simplification is the process of reducing expressions to simpler or more concise forms. It's about understanding how different algebraic elements interact and transforming the expression without changing its value. In our given exercise, we're asked to simplify \( (x^{1/2} + y^{1/2})(x^{1/2} - y^{1/2}) \).
The simplification process often involves identifying patterns or structures within the expression that match well-known formulas.

• In this exercise, the formula being used is the difference of squares. This allows us to simplify without expanding each term individually.
• The difference of squares formula, \((a+b)(a-b) = a^2 - b^2\), simplifies the expression by considering \(a = x^{1/2}\) and \(b = y^{1/2}\).

By applying this formula, we quickly transform the expression into a simpler form:
\( (x^{1/2})^2 - (y^{1/2})^2 \), which simplifies further to \( x - y \).

Radicals

Radicals are mathematical expressions that involve roots, like square roots or cube roots. In this exercise, we encounter radicals in the form of \( x^{1/2} \) and \( y^{1/2} \).
Understanding how radicals work is essential for solving and simplifying expressions that contain them.
Radicals can be simplified by recognizing their properties:

• The square root of a squared number returns the original number — essentially, the square root operation \( (x^{1/2})^2 \) returns \( x \).
• When radicals appear in expressions, looking for opportunities to apply simplification formulas like the difference of squares can make the process more manageable.

In our scenario, simplifying \( (x^{1/2})^2 \) allows us to remove the radical and simplify directly to \( x \).
Likewise, \( (y^{1/2})^2 \) simplifies to \( y \), giving us the final result of \( x - y \).

Polynomial Expressions

Polynomial expressions in algebra are mathematical expressions that involve sums, products, and powers of variables. They can vary in complexity and form, involving both whole numbers and fractions, as well as variables raised to various powers. In this exercise, the expression
\((x^{1/2} + y^{1/2})(x^{1/2} - y^{1/2}) \) can initially seem complex due to the presence of square roots.

• By identifying this expression as a special type of polynomial — a binomial — that fits the difference of squares pattern, we can simplify it significantly.
• The resulting expression \( x - y \) is a linear polynomial, representing a direct subtraction without any roots or powers.

Polynomials are foundational in algebra, underlining the importance of recognizing patterns and forms like the difference of squares.
Such recognition simplifies the task of manipulating and solving polynomial expressions.