Q. 242 - Intermediate Algebra

Question

Explain the process for determining whether two radicals are like or unlike. Make sure your answer makes sense for radicals containing both numbers and variables.

Step-by-Step Solution

Answer

Like radicals: 

Like radicals are radical expressions with the same index and the same radicand.

Unlike radicals:

If the radical expression have either different index or different radicand, it is called unlike radicals and it cannot be combined.

Step 1. To explain

To explain the process for determining whether two radicals are like or unlike.

Step 2. Explanation for like and unlike radicals

Like radicals: 

Like radicals are radical expressions with the same index and the same radicand.

Unlike radicals:

If the radical expression have either different index or different radicand, it is called unlike radicals and it cannot be combined.

Step 3. Example for like radicals

Consider an example, $5\sqrt{y}+7\sqrt{y}$,

Here the radicals have same index and same radicals, so it is like radical and we can perform addition operation.

$5\sqrt{y}+7\sqrt{y}=12\sqrt{y}$

Consider $2\sqrt{2}+8\sqrt{2}$

Since they are like radicals, just add terms,

$2\sqrt{2}+8\sqrt{2}=10\sqrt{2}$

Step 4. Example for unlike radicals

Consider an example, $x\sqrt[3]{y}+y^3\sqrt{x}$

Here the radicals have different index and radicals, so it is unlike radical and we cannot perform addition operation.

$x\sqrt[3]{y}+y^3\sqrt{x}$ cannot be reduced further.

Consider $5\sqrt{3}+5\sqrt{2}$

Since it is unlike, it cannot be combined,

$5\sqrt{3}+5\sqrt{2}=5(\sqrt{3}+\sqrt{2})$