Multiply. See Example 7.(√2 + 1) (√3 + 1)

Verified step by step guidance

1. Identify the expression to be multiplied: \((\sqrt{2} + 1)(\sqrt{3} + 1)\).
2. Apply the distributive property (also known as the FOIL method for binomials): \((a + b)(c + d) = ac + ad + bc + bd\).
3. Multiply the first terms: \(\sqrt{2} \times \sqrt{3}\).
4. Multiply the outer terms: \(\sqrt{2} \times 1\).
5. Multiply the inner terms: \(1 \times \sqrt{3}\) and the last terms: \(1 \times 1\).

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. For example, in (a + b)(c + d), you would calculate ac, ad, bc, and bd.

Square Roots

Square roots are values that, when multiplied by themselves, yield the original number. In the context of the expression (√2 + 1)(√3 + 1), understanding how to manipulate square roots is essential. For instance, √2 and √3 are irrational numbers, and their properties can affect the simplification of the expression when combined with integers.

Simplifying Radical Expressions

Simplifying radical expressions involves reducing the expression to its simplest form, which may include combining like terms or rationalizing denominators. In the case of the product (√2 + 1)(√3 + 1), after multiplication, you may encounter terms that can be simplified further, such as combining square roots or simplifying coefficients to achieve a cleaner expression.