Perform the indicated operations. Assume all variables represent positive real numbers. (√3 + √8)²

Verified step by step guidance

1. Recognize that the expression is a square of a binomial: \((\sqrt{3} + \sqrt{8})^2\).
2. Recall the formula for the square of a binomial: \((a + b)^2 = a^2 + 2ab + b^2\).
3. Identify \(a = \sqrt{3}\) and \(b = \sqrt{8}\), then substitute into the formula: \((\sqrt{3})^2 + 2 \times \sqrt{3} \times \sqrt{8} + (\sqrt{8})^2\).
4. Simplify each term: \((\sqrt{3})^2 = 3\) and \((\sqrt{8})^2 = 8\), so the expression becomes \$3 + 2 \times \sqrt{3} \times \sqrt{8} + 8$.
5. Combine the constants and simplify the middle term by multiplying the square roots: \$3 + 8 + 2 \times \sqrt{3 \times 8}$.

Key Concepts

Simplifying Radicals

Simplifying radicals involves rewriting square roots in their simplest form by factoring out perfect squares. For example, √8 can be simplified to 2√2 because 8 = 4 × 2 and √4 = 2. This step makes further operations easier and clearer.

Binomial Expansion (Square of a Sum)

The square of a sum (a + b)² expands to a² + 2ab + b². This formula allows you to multiply expressions like (√3 + √8)² without directly multiplying the binomial twice, simplifying the calculation process.

Multiplying Radicals

When multiplying radicals, multiply the numbers inside the square roots together, then simplify if possible. For example, √3 × √8 = √(3×8) = √24, which can be further simplified. This is essential for calculating the middle term in the binomial expansion.