Answer :
To determine which of the given options is true when adding or subtracting radicals, let's carefully analyze each option:
1. Determine if the roots or index are the same:
- This is essential because radicals can only be added or subtracted if they have the same index and the same radicand (the number inside the radical sign). For example, [tex]\(\sqrt{2} + \sqrt{2}\)[/tex] can be simplified, but [tex]\(\sqrt{2} + \sqrt{3}\)[/tex] cannot be combined.
2. If terms are the same, combine, then add or subtract the coefficients:
- Once you have confirmed that the indices and radicands are the same, you can combine them by adding or subtracting the coefficients. For example, [tex]\(3\sqrt{5} + 2\sqrt{5} = (3 + 2)\sqrt{5} = 5\sqrt{5}\)[/tex].
3. You cannot add or subtract if roots are not the same:
- This statement reinforces the first rule. If the indices or radicands are different, you cannot combine the radicals through addition or subtraction. For instance, [tex]\(\sqrt{7} + \sqrt{3}\)[/tex] cannot be simplified further since the radicands are different.
4. All of the above:
- This option states that all the previous statements are true.
Given the explanations above, it's clear that all three individual statements about adding or subtracting radicals are accurate. Therefore, the correct choice is:
Option 4 - All of the above
1. Determine if the roots or index are the same:
- This is essential because radicals can only be added or subtracted if they have the same index and the same radicand (the number inside the radical sign). For example, [tex]\(\sqrt{2} + \sqrt{2}\)[/tex] can be simplified, but [tex]\(\sqrt{2} + \sqrt{3}\)[/tex] cannot be combined.
2. If terms are the same, combine, then add or subtract the coefficients:
- Once you have confirmed that the indices and radicands are the same, you can combine them by adding or subtracting the coefficients. For example, [tex]\(3\sqrt{5} + 2\sqrt{5} = (3 + 2)\sqrt{5} = 5\sqrt{5}\)[/tex].
3. You cannot add or subtract if roots are not the same:
- This statement reinforces the first rule. If the indices or radicands are different, you cannot combine the radicals through addition or subtraction. For instance, [tex]\(\sqrt{7} + \sqrt{3}\)[/tex] cannot be simplified further since the radicands are different.
4. All of the above:
- This option states that all the previous statements are true.
Given the explanations above, it's clear that all three individual statements about adding or subtracting radicals are accurate. Therefore, the correct choice is:
Option 4 - All of the above
