Question

When can you add or subtract radicals without having to simplify?

(a) When the radicals have the same radicand (the number under the radical symbol).
(b) When the radicals have the same index (the number next to the radical symbol).
(c) When the radicals have the same sign (positive or negative).
(d) When the radicals are both perfect squares.

Answer 1

You can add or subtract radicals directly when they have the same radicand and the same index, as they are like terms in this case.

Step-by-step explanation:

You can add or subtract radicals without having to simplify when the radicals have the same radicand and the same index. This is because radicals can only be combined in addition or subtraction when they are like terms, similar to how one would combine similar algebraic terms. For instance, √2 + √2 is permissible because both terms have the radical √ and the radicand 2. However, √3 + √2 cannot be simplified through addition or subtraction because the radicands 2 and 3 are different.

Let's clarify with an example: if we have 3√2 + 2√2, since both radicals have the same index (2 being an implied square root) and the same radicand (2), we can add them to get 5√2. However, signs and whether the terms are perfect squares do not affect our ability to add or subtract radicals.

Answer 2

Answer: Choice A

When the radicals have the same radicand (the number under the radical symbol).

Example:
`3√(2)+5√(2) = 8√(2)` since we add the coefficients out front (3+5 = 8) while leaving the square root terms as they are. Think of it like saying 3x+5x = 8x where
`\text{x} = √(2)`

-------------

If the radicands weren't the same, then we'd have to do a bit of simplification as shown in the scratch work below.

`√(75)+√(48)\\\\=√(25*3)+√(16*3)\\\\=√(25)*√(3)+√(16)*√(3)\\\\=5√(3)+4√(3)\\\\=9√(3)\\\\`

In short,

`√(75)+√(48)=9√(3)\\\\`

In this example, I factored each radicand so that I pulled out the largest perfect square factor possible. That leaves behind 3 under the square root. From that point, follow the same idea as the previous section to combine like terms.

Notes:

• You may not be able to simplify some square root expressions. For instance
  `√(37)` cannot be simplified as there are no perfect square factors (other than 1).
• Even if you are able to simplify square root expressions, there may not be like terms. Example:
  `√(75)+√(8)=5√(3)+2√(2)\\\\` which has no like terms. That second expression is of the form 5x+2y where
  `\text{x}=√(3) \text{ and } \text{y} = √(2)`