Ch. R - Review of Basic Concepts

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. R - Review of Basic Concepts

Problem 123

Chapter 1, Problem 123

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

Verified step by step guidance

Recognize that the expression is a square of a binomial: \((\sqrt{3} + \sqrt{8})^2\).

Recall the formula for the square of a binomial: \((a + b)^2 = a^2 + 2ab + b^2\).

Identify \(a = \sqrt{3}\) and \(b = \sqrt{8}\), then substitute into the formula: \((\sqrt{3})^2 + 2 \times \sqrt{3} \times \sqrt{8} + (\sqrt{8})^2\).

Simplify each term: \((\sqrt{3})^2 = 3\) and \((\sqrt{8})^2 = 8\), so the expression becomes \(3 + 2 \times \sqrt{3} \times \sqrt{8} + 8\).

Combine the constants and simplify the middle term by multiplying the square roots: \(3 + 8 + 2 \times \sqrt{3 \times 8}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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