Problem 46 - Algebra for College Students

Question

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. \((2 \sqrt{3}+\sqrt{11})(2 \sqrt{3}-\sqrt{11})\)

Step-by-Step Solution

Answer

The product is 1.

Step 1: Identify the Pattern

The expression \((2 \sqrt{3} + \sqrt{11})(2 \sqrt{3} - \sqrt{11})\) fits the pattern of a difference of squares: \((a+b)(a-b) = a^2 - b^2\). Here, \(a = 2\sqrt{3}\) and \(b = \sqrt{11}\).

Step 2: Square the Terms

Calculate \(a^2\). Since \(a = 2\sqrt{3}\), we have \(a^2 = (2\sqrt{3})^2 = 4 \cdot 3 = 12\). Similarly, calculate \(b^2\). Since \(b = \sqrt{11}\), we have \(b^2 = (\sqrt{11})^2 = 11\).

Step 3: Apply the Difference of Squares Formula

Substitute the squared terms into the formula: \(a^2 - b^2 = 12 - 11\).

Step 4: Simplify the Result

Calculate \(12 - 11 = 1\). The product \((2\sqrt{3} + \sqrt{11})(2\sqrt{3} - \sqrt{11})\) simplifies to 1.

Key Concepts

Simplest Radical Form

The simplest radical form of a number is its most simplified form involving square roots. When working with radicals, it's crucial to break down the expression such that no radicals exist in the denominator and there are no perfect square factors left under the radical sign other than 1.
For example:

• The simplest radical form of \( \sqrt{50} \) is \( 5\sqrt{2} \), because \( 50 = 25 \times 2 \) and \( 25 \) is a perfect square.


• Once you express a number in terms of its prime factors, you can take the square root of the pairs.

Expressing results in the simplest radical form helps make calculations more manageable and expressions more straightforward.

Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. The square root symbol (\( \sqrt{} \)) is used to denote this operation. For instance, the square root of 9 is 3, because \( 3 \times 3 = 9 \).

• Square roots can be either positive or negative, but by definition, the square root function returns the nonnegative root, known as the principal square root.


• Every positive number has two square roots: one positive (principal root) and one negative, but only the principal root is considered when dealing with real numbers.

Understanding square roots is fundamental to simplifying radical expressions and performing operations like finding the simplest radical form.

Nonnegative Real Numbers

Nonnegative real numbers include all the positive numbers and zero. They are an essential part of mathematical calculations, especially when dealing with operations like taking square roots.

• Variables representing nonnegative real numbers ensure that expressions under a square root remain non-negative, which is critical since the square root of a negative number isn't defined in the set of real numbers.


• In mathematical problems, assuming that variables are nonnegative simplifies operations and eliminates the need for considering complex numbers.

By restricting variables to nonnegative real numbers, we avoid the complications that arise in radical expressions and ensure the results are within the scope of real number arithmetic.