Find each product. Assume all variables represent positive real numbers. (x+x^1/2)(x-x^1/2)

Verified step by step guidance

1. Recognize that the expression \((x + x^{1/2})(x - x^{1/2})\) is in the form of a product of conjugates, which follows the pattern \((a + b)(a - b) = a^2 - b^2\).
2. Identify \(a = x\) and \(b = x^{1/2}\) in the given expression.
3. Apply the difference of squares formula: \(a^2 - b^2 = x^2 - (x^{1/2})^2\).
4. Simplify the term \((x^{1/2})^2\) by using the property of exponents \((x^{m})^{n} = x^{m \times n}\), which gives \(x^{1/2 \times 2} = x^1 = x\).
5. Write the simplified expression as \(x^2 - x\), which is the product of the original expression.

Key Concepts

Properties of Exponents

Understanding how to manipulate exponents is essential, especially when dealing with fractional exponents like x^(1/2). This concept includes rules such as multiplying powers with the same base by adding exponents and simplifying expressions involving roots and powers.

Difference of Squares

The expression (a + b)(a - b) equals a^2 - b^2, a fundamental algebraic identity. Recognizing this pattern allows for quick simplification of products involving sums and differences of the same terms, which is crucial for efficiently solving the given problem.

Polynomial Multiplication

Multiplying polynomials involves distributing each term in the first polynomial to every term in the second. Mastery of this process ensures accurate expansion of expressions like (x + x^(1/2))(x - x^(1/2)) before simplifying.