Perform the indicated operations. Assume all variables represent positive real numbers. 5√6 + 2√10

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Identify the terms in the expression: $5\sqrt{6} + 2\sqrt{10}$. Notice that these are two separate radical terms.

Check if the radicals have the same radicand (the number inside the square root). Here, one term has $\sqrt{6}$ and the other has $\sqrt{10}$, which are different.

Since the radicands are different, these terms are not like terms and cannot be combined by addition or subtraction directly.

Consider if either radical can be simplified by factoring out perfect squares. For $\sqrt{6}$ and $\sqrt{10}$, check their prime factorizations: 6 = 2 × 3 and 10 = 2 × 5. Neither contains a perfect square factor other than 1.

Since the radicals cannot be simplified further and are unlike terms, the expression $5\sqrt{6} + 2\sqrt{10}$ is already in its simplest form.

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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 expressing a square root in its simplest form by factoring out perfect squares. For example, √50 can be simplified to 5√2 because 50 = 25 × 2 and √25 = 5. This process helps in combining or comparing radical expressions.

Like Radicals

Like radicals have the same radicand (the number inside the square root) and index. Only like radicals can be added or subtracted directly by combining their coefficients. For example, 3√2 + 5√2 = 8√2, but 3√2 + 5√3 cannot be combined directly.

Addition of Radical Expressions

To add radical expressions, first simplify each radical and identify like radicals. Then, add the coefficients of the like radicals while keeping the radical part unchanged. If radicals are unlike, the expression remains as a sum of separate terms.

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