Like Radicals
Combine like radicals in Grade 10 algebra. Identify matching radicand and index, add or subtract coefficients of like radicals, just as you combine like terms in polynomial algebra.
Key Concepts
When radical expressions have the same radicand and index, they are like radicals and can be combined. This is analogous to combining like terms, such as $2x + 3x = 5x$.
$4\sqrt{5} + 9\sqrt{5} = (4+9)\sqrt{5} = 13\sqrt{5}$ $12\sqrt{7} 3\sqrt{7} 2\sqrt{7} = (12 3 2)\sqrt{7} = 7\sqrt{7}$ $\sqrt{18} + \sqrt{50} = \sqrt{9 \cdot 2} + \sqrt{25 \cdot 2} = 3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$.
Think of radicals like items with the same name. You can combine '3 apples' and '5 apples' to get '8 apples'. In math, you can combine $3\sqrt{2}$ and $5\sqrt{2}$ to get $8\sqrt{2}$. If the number and type of root are different (like $\sqrt{2}$ and $\sqrt{3}$), they're not 'like' and can't be combined!
Common Questions
What are like radicals?
Like radicals have the same radicand (number under the radical) and the same index (root degree). For example, 2√3 and 5√3 are like radicals; 2√3 and 2√5 are not.
How do you combine like radicals?
Add or subtract the coefficients just like combining like terms. 2√3 + 5√3 = 7√3. The radical part stays the same — only the coefficients change.
Can unlike radicals be combined directly?
No. Unlike radicals (different radicands or different indices) cannot be directly combined. First simplify each radical — sometimes simplification reveals like radicals that can then be combined.