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Understanding square roots is crucial in solving radical expressions. A square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 36 is 6, because \(6 \times 6 = 36\). Recognizing this concept helps in evaluating expressions involving radicals.
To evaluate \(\sqrt{b^{2}+10a}\), first find the value inside the square root, then determine its square root. The expression under the square root must be simplified to a perfect square if possible, allowing easy extraction of the square root value. This involves breaking down the number's components and checking if they multiply back to the number inside the radical.

Substitution is a method of replacing variables with their known values in an equation or expression. This process helps simplify the expression to a manageable mathematical operation.
For example, in the expression \(\sqrt{b^{2}+10a}\), knowing that \(a=2\) and \(b=4\), you substitute these values into the expression. Substitution transforms \(\sqrt{b^{2}+10a}\) into \(\sqrt{4^{2}+10(2)}\). This step is essential because it allows one to work with numerical values instead of variables, simplifying the expression further.
Always ensure that the variables are replaced correctly and double-check the calculations, especially with variables raised to a power like \(b^{2}\). This prevents potential calculation errors that might lead to an incorrect final answer.

Simplification involves reducing an expression to its simplest form. This step makes it easier to interpret and calculate the final result. With the given expression \(\sqrt{4^{2}+10(2)}\), you simplify the terms inside the square root first by calculating \(4^{2}\) to give 16, and evaluating \(10(2)\) to get 20.
The next step is to add these numbers: \(16+20 = 36\). Now, the expression inside the square root is simply \(\sqrt{36}\).

Simplification is critical as it clearly presents the most straightforward form of the problem, making it easy to apply mathematical operations such as taking the square root. By understanding simplification, you ensure that each step leads you closer to finding an accurate answer.