Basic operations with surds

A square root can be simplified if the number under the radical has a square factor. 45 = 9 * 5, and 9 is a square number, so √45 = 3√5.

Radicals can only be added if they have the same radicand (the number inside the root). Since √2 and √3 are different, they cannot be combined.

Since both terms contain √2, they are like radicals. Add the coefficients: 2 + 5 = 7, resulting in 7√2.

A square factor is a perfect square (other than 1) that divides the number. 89 is a prime number and has no square factors.

A square factor is a perfect square that divides the number. 90 is divisible by 9 (which is 3^2), so it has a square factor.

A square factor is a perfect square that divides the number. 45 is divisible by 9 (which is 3^2), so it has a square factor.

To simplify √80, find the largest square factor. 80 = 16 * 5. Thus, √80 = √16 * √5 = 4√5.

To simplify √50, factor it into a square and another number: 50 = 25 * 2. Then √50 = √25 * √2 = 5√2. Option A correctly identifies the prime factorization logic.

Factor 28 into a square and another number: 28 = 4 * 7. Then √28 = √4 * √7 = 2√7.

To move the coefficient 3 inside the radical, square it: 3√2 = √(3^2 * 2) = √(9 * 2) = √18.

Using the property √a * √b = √(a * b), √18 can be split into factors like √9 * √2.

A square number is the result of an integer multiplied by itself (e.g., 0, 1, 4, 9, 16, 25, 36, 49, 64). 24 is not a perfect square.

The expression 2√3 simply means 2 multiplied by √3. This is the definition of the coefficient notation.

Treat √2 as a variable (like x). So, x + x = 2x. Therefore, √2 + √2 = 2√2.

Using the rule √a * √a = a, √2 * √2 = 2. However, √2 * √2 is also equal to √4, which simplifies to 2. Option A is technically correct as √4 = 2.