Add or subtract, as indicated. See Example 6.5√3 + √12

Verified step by step guidance

1. Identify the terms: \(5\sqrt{3}\) and \(\sqrt{12}\).
2. Simplify \(\sqrt{12}\) by finding its prime factors: \(12 = 4 \times 3 = 2^2 \times 3\).
3. Rewrite \(\sqrt{12}\) as \(\sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}\).
4. Now, the expression becomes \(5\sqrt{3} + 2\sqrt{3}\).
5. Combine like terms: \((5 + 2)\sqrt{3}\).

Key Concepts

Radical Simplification

Radical simplification involves reducing square roots to their simplest form. For example, √12 can be simplified by factoring it into √(4*3), which equals 2√3. This process is essential for combining like terms in expressions involving square roots.

Like Terms

Like terms are terms that contain the same variable raised to the same power or, in the case of radicals, the same root. In the expression 5√3 + √12, after simplifying √12 to 2√3, we can combine 5√3 and 2√3 to get 7√3, demonstrating the importance of identifying and combining like terms.

Addition of Radicals

When adding radicals, it is crucial to ensure that the terms being added are like terms. This means they must have the same radicand (the number under the square root). For instance, in the expression 5√3 + 2√3, both terms are multiples of √3, allowing us to add their coefficients to arrive at a final result of 7√3.