Problem 36 - Algebra for College Students

Question

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. \((\sqrt{2}+\sqrt{3})(\sqrt{5}-\sqrt{7})\)

Step-by-Step Solution

Answer

\(\sqrt{10} - \sqrt{14} + \sqrt{15} - \sqrt{21}\)

Step 1: Use the Distributive Property

To simplify the product \((\sqrt{2}+\sqrt{3})(\sqrt{5}-\sqrt{7})\), apply the distributive property, also known as the FOIL method, which stands for First, Outer, Inner, and Last. Calculate each product as follows:1. First: \(\sqrt{2} \cdot \sqrt{5} = \sqrt{10}\).2. Outer: \(\sqrt{2} \cdot (-\sqrt{7}) = -\sqrt{14}\).3. Inner: \(\sqrt{3} \cdot \sqrt{5} = \sqrt{15}\).4. Last: \(\sqrt{3} \cdot (-\sqrt{7}) = -\sqrt{21}\).

Step 2: Combine Like Terms

After distributing, combine all the terms obtained:\[ \sqrt{10} - \sqrt{14} + \sqrt{15} - \sqrt{21}. \]

Step 3: Simplify the Expression

The expression \(\sqrt{10} - \sqrt{14} + \sqrt{15} - \sqrt{21}\) is already in its simplest radical form as there are no like terms to combine further and each of the radicals is in simplest form.

Key Concepts

Simplifying Radicals

Radicals, often known as roots, express numbers that can be represented as a root of another number. The most common radical is the square root. Simplifying radicals means finding an equivalent expression where the radicand (the number under the radical symbol) is as simple as possible.
For example,

• The square root of 18 can be simplified to \( 3\sqrt{2} \) because 18 is equal to \( 9 \times 2 \), and the square root of 9 is 3.
• Radicals like \( \sqrt{2} \) or \( \sqrt{5} \) are already in their simplest form if the number under the root sign is a prime number or cannot be broken down further into smaller square factors.

When simplifying a radical completely, try to identify perfect squares within the radicand, and factor them out accordingly. This reduces the radicand while keeping the radical expression equivalent.

Distributive Property

The Distributive Property is a fundamental rule of algebra. It allows a term to be distributed across terms inside brackets, making multiplication across a sum or difference possible. In essence, you're breaking up the expression into smaller parts that are easier to handle individually. In radical expressions, like \((\sqrt{2}+\sqrt{3})(\sqrt{5}-\sqrt{7})\), you can apply this property to each term:

• Multiply the first term of each binomial.
• Then proceed to distribute each remaining term in the first binomial with each term in the second binomial.

Using the distributive property helps simplify complex expressions and prepares the expression for combining like terms. It's essential when you're dealing with binomials involving radicals or any form of polynomials.

FOIL Method

The FOIL Method is a specific application of the distributive property when dealing with two binomials. FOIL, an acronym for First, Outer, Inner, Last, structures the order in which you multiply terms to ensure none are missed. Here's how you apply it for radical expressions:

• First: Multiply the first terms in each binomial, e.g., \( \sqrt{2} \cdot \sqrt{5} \).
• Outer: Multiply the outer terms, e.g., \( \sqrt{2} \cdot (-\sqrt{7}) \).
• Inner: Multiply the inner terms, e.g., \( \sqrt{3} \cdot \sqrt{5} \).
• Last: Multiply the last terms in each binomial, e.g., \( \sqrt{3} \cdot (-\sqrt{7}) \).

The method ensures all parts of the expression are accounted for. Once calculated, these products can be combined or simplified further. The FOIL method is particularly useful in handling binomial expressions with radicals because it provides a systematic process for expanding expressions.