Identify two key differences between the procedures for multiplying radicals and procedures for adding and subtracting them

Answer 1

Here are two key differences between the procedures for multiplying radicals and the procedures for adding and subtracting them:

1. Combining versus Simplifying:

   • Addition/Subtraction: When adding or subtracting radicals, you can only combine them if they have the same radicand (the number inside the radical). For example, $\sqrt{2}+3\sqrt{2}=4\sqrt{2}$ because they have the same radicand. However, $\sqrt{2}+\sqrt{3}$ cannot be combined further and remains $\sqrt{2}+\sqrt{3}$.
   • Multiplication: When multiplying radicals, you can simply multiply the radicands regardless of whether they are the same or not. For example, $\sqrt{2}\times \sqrt{3}=\sqrt{6}$. The multiplication of radicals follows the property $\sqrt{a}\times \sqrt{b}=\sqrt{a\cdot b}$.

2. Distribution versus Factorization:

   • Addition/Subtraction: In addition or subtraction, the radicals are treated as terms that can be rearranged, but their individual components remain separated. Combining them often requires factoring out common terms, like factoring a $\sqrt{2}$ out from $\sqrt{2}+3\sqrt{2}$.
   • Multiplication: In multiplication, you apply the distributive property, which often involves changing the form of the radicals. For example, $(\sqrt{2}+\sqrt{3})(\sqrt{2}-\sqrt{3})$ results in a simplified form via the difference of squares: $2-3=-1$.

These key differences highlight the distinct approaches needed for handling operations with radicals.