In mathematics, surds are irrational numbers expressed in the form of square roots (√) or other root symbols. When comparing or equating surds, the concept of equality of surds is important.
 
Two surds are considered equal if and only if their radicands (the numbers inside the square root symbol) are equal. In other words, if two surds have the same radicand, they are equal. However, it’s important to note that the surds should also have the same index (root) for them to be considered equal.
 
For example:
√2 and √2 are equal because they have the same radicand (2) and the same index (2).
 
√3 and √5 are not equal because they have different radicands (3 and 5) even though they have the same index (2).
 
To further illustrate the concept of equality of surds, consider the following examples:
 
Example 1:
√8 = √(4 × 2) = √4 × √2 = 2√2
 
In this example, √8 is equal to 2√2 because both expressions simplify to the same value.
 
Example 2:
√(12 + 5) = √17
 
In this example, the expression inside the square root simplifies to 17, so √(12 + 5) is equal to √17.
 
It’s important to simplify surds as much as possible to determine their equality. This involves finding perfect square factors of the radicand and simplifying them.
 
In summary, the equality of surds is determined by comparing their radicands and indices. If the radicands are the same and the indices are the same, the surds are considered equal.