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

Verified step by step guidance

1. Identify the expression to multiply: \((\sqrt{2} + 1)(\sqrt{3} + 1)\).
2. Apply the distributive property (also known as FOIL for binomials): multiply each term in the first parenthesis by each term in the second parenthesis.
3. Write out the multiplication explicitly: \((\sqrt{2} \times \sqrt{3}) + (\sqrt{2} \times 1) + (1 \times \sqrt{3}) + (1 \times 1)\).
4. Simplify each term: \(\sqrt{2} \times \sqrt{3} = \sqrt{6}\), \(\sqrt{2} \times 1 = \sqrt{2}\), \(1 \times \sqrt{3} = \sqrt{3}\), and \(1 \times 1 = 1\).
5. Combine all simplified terms to write the expanded expression: \(\sqrt{6} + \sqrt{2} + \sqrt{3} + 1\).

Key Concepts

Binomial Multiplication

Multiplying two binomials involves applying the distributive property (also known as FOIL: First, Outer, Inner, Last) to each term in the first binomial with each term in the second. This process expands the product into a sum of terms.

Simplification of Radicals

After multiplication, terms involving square roots (radicals) may appear. Simplifying radicals involves combining like terms and reducing square roots when possible, such as recognizing that √a * √b = √(a*b).

Combining Like Terms

Once the product is expanded, terms that are similar (e.g., constants or terms with the same radical) should be combined to simplify the expression into its simplest form.