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 the expression as a difference of squares: \((a+b)(a-b) = a^2 - b^2\).
2. Identify \(a = x\) and \(b = x^{1/2}\).
3. Apply the difference of squares formula: \((x + x^{1/2})(x - x^{1/2}) = x^2 - (x^{1/2})^2\).
4. Calculate \((x^{1/2})^2\) which is \(x^{1/2 \times 2} = x^1 = x\).
5. Substitute back into the expression: \(x^2 - x\).

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves distributing each term in one polynomial to every term in another polynomial. In this case, we apply the distributive property to the expression (x + x^(1/2))(x - x^(1/2)), which requires careful attention to the signs and exponents of the terms involved.

Exponents and Radicals

Understanding exponents and radicals is crucial for manipulating expressions involving powers. In the given expression, x^(1/2) represents the square root of x, and recognizing how to combine and simplify terms with different exponents is essential for finding the product accurately.

Factoring and Simplifying Expressions

After multiplying the polynomials, the resulting expression may often be simplified or factored further. This involves combining like terms and applying algebraic identities, which can help in expressing the product in a more manageable form, making it easier to analyze or solve.