Add or subtract, as indicated. See Example 6. 5√3 + √12

Verified step by step guidance

1. Identify the terms to be added: \(5\sqrt{3}\) and \(\sqrt{12}\).
2. Simplify the square root in the second term: \(\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}\).
3. Rewrite the expression with the simplified term: \(5\sqrt{3} + 2\sqrt{3}\).
4. Since both terms have the same radical part (\(\sqrt{3}\)), combine the coefficients: \((5 + 2)\sqrt{3}\).
5. Express the final simplified form as \(7\sqrt{3}\) (do not calculate the decimal value).

Key Concepts

Simplifying Radicals

Simplifying radicals involves expressing a square root in its simplest form by factoring out perfect squares. For example, √12 can be simplified to 2√3 because 12 = 4 × 3 and √4 = 2. This step is essential before performing addition or subtraction of radicals.

Like Radicals

Like radicals have the same radicand (the number inside the square root). Only like radicals can be added or subtracted directly by combining their coefficients. For instance, 5√3 and 2√3 are like radicals and can be combined as (5 + 2)√3 = 7√3.

Adding and Subtracting Radicals

To add or subtract radicals, first simplify them and ensure they are like radicals. Then, add or subtract their coefficients while keeping the radical part unchanged. This process is similar to combining like terms in algebra.