In Exercises 1–38, multiply as indicated. If possible, simplify any radical expressions that appear in the product.(7 - 2√7) (5 - 3√7)

Verified step by step guidance

1. Identify the expression to be multiplied: \((7 - 2\sqrt{7})(5 - 3\sqrt{7})\).
2. Apply the distributive property (also known as the FOIL method for binomials): First, multiply the first terms: \(7 \times 5\).
3. Next, multiply the outer terms: \(7 \times -3\sqrt{7}\).
4. Then, multiply the inner terms: \(-2\sqrt{7} \times 5\).
5. Finally, multiply the last terms: \(-2\sqrt{7} \times -3\sqrt{7}\).

Key Concepts

Multiplication of Binomials

Multiplying binomials involves applying the distributive property, often referred to as the FOIL method (First, Outside, Inside, Last). This technique helps in systematically multiplying each term in the first binomial by each term in the second binomial, ensuring that all combinations are accounted for in the final expression.

Radical Expressions

Radical expressions contain roots, such as square roots, cube roots, etc. Simplifying these expressions often involves factoring out perfect squares or cubes, allowing for a more manageable form. Understanding how to manipulate radicals is essential for simplifying the results of operations involving them.

Simplification of Expressions

Simplification involves reducing an expression to its simplest form, which may include combining like terms, reducing fractions, or simplifying radicals. This process is crucial in algebra as it makes expressions easier to work with and understand, especially when preparing for further calculations or evaluations.