Problem 48 - Algebra for College Students

Question

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. \((2 \sqrt{x}-5 \sqrt{y})(2 \sqrt{x}+5 \sqrt{y})\)

Step-by-Step Solution

Answer

The simplified product is \(4x - 25y\).

Step 1: Identify the expression pattern

The expression \((2 \sqrt{x}-5 \sqrt{y})(2 \sqrt{x}+5 \sqrt{y})\)fits the pattern of a difference of squares formula, \((a - b)(a + b) = a^2 - b^2\). Here, \(a = 2 \sqrt{x}\) and \(b = 5 \sqrt{y}\).

Step 2: Apply the difference of squares formula

Using the difference of squares formula, calculate \(a^2 - b^2\).First, find \(a^2 = (2 \sqrt{x})^2 = 4x\) and \(b^2 = (5 \sqrt{y})^2 = 25y\). Then, subtract the two squares to get \(4x - 25y\).

Step 3: Write the final simplified expression

The product of the expression \((2 \sqrt{x}-5 \sqrt{y})(2 \sqrt{x}+5 \sqrt{y})\)simplified is \(4x - 25y\).

Key Concepts

Simplest Radical Form

Finding the simplest radical form is an important step in working with expressions that involve square roots or other radicals. This means writing a radical expression in its most simplified version without changing its value. To achieve this:

• Ensure that there are no perfect square factors remaining under any square roots.
• Reduce each radical such that no integral factor is left inside the radical except primes.

For example, in the given expression \(2 \sqrt{x}-5 \sqrt{y})(2 \sqrt{x}+5 \sqrt{y})\), the square roots are already expressed in their simplest form. However, once the operation is performed, we ensure that the result is left in a form, like \(4x - 25y\), which contains no further simplifiable radicals.

Simplifying Expressions

Simplifying expressions is a crucial part of working with algebraic equations. It involves combining like terms and reducing the expression to its simplest form. In our exercise, we simplified by recognizing patterns:

• Identify if the expression matches any known algebraic identities, such as the difference of squares.
• Convert complex expressions into simpler forms by applying algebraic rules, like squaring numbers and canceling out terms where possible.

For \(2 \sqrt{x}-5 \sqrt{y})(2 \sqrt{x}+5 \sqrt{y})\), the simplification process involved applying the difference of squares, leading us directly to \(4x - 25y\). Here we ensure that the expression cannot be reduced any further and that it is presented as concisely as possible.

Algebraic Products

An algebraic product in mathematics is the result of multiplying two or more expressions together. Understanding how to manage and manipulate these products is essential in algebra. To successfully handle algebraic products:

• Know and use key formulas, such as the difference of squares, which is \(a^2 - b^2\) when you have \(a - b\) and \(a + b\).
• Recognize the individual components within expressions that you can apply these formulas to, such as taking \(a = 2 \sqrt{x}\) and \(b = 5 \sqrt{y}\) in our example.

In this problem, the algebraic product forms a perfect opportunity to use the difference of squares formula and simplify the operation right to the solution: \(4x - 25y\). By understanding this concept, you can work through even more complex multiplications in algebra.