To add and subtract radical expressions, you must identify "like" radicals. This process is identical to combining like terms in polynomial algebra.
Need to review the basics? Start with our introduction to radicals.
Definition of Like Radicals
Radical expressions are considered like only if they possess the exact same index and the same radicand.
- \( 6 \sqrt[3]{5} \) and \( -5 \sqrt[3]{5} \) are like (Index: 3, Radicand: 5).
- \( 7 \sqrt[3]{8} \) and \( -5 \sqrt[3]{9} \) are not like (Radicands 8 and 9 differ).
- \( 3 \sqrt{2x} \) and \( -5 \sqrt{2x} \) are like (Index: 2, Radicand: \(2x\)).
Basic Addition & Subtraction Examples
Only like radicals may be added or subtracted by factoring out the common radical and combining their coefficients.
Example 1: \( 4 \sqrt[3]{5} + 7 \sqrt[3]{5} \)
Example 2: \( 9 \sqrt{13} - 11 \sqrt{13} \)
Example 3: \( -8 \sqrt[4]{2x+1} + 6 \sqrt[4]{2x+1} \)
Example 4: \( -\sqrt{2xy} - 4 \sqrt{2xy} + 23 \sqrt{2xy} \)
Transforming Unlike Radicals
When radicands differ, simplify them first by factoring out perfect squares (or cubes) to see if they can be converted into like radicals.
Example 5: \( 4\sqrt{8} - 6\sqrt{2} \)
Example 6: \( -4\sqrt{12} + 12\sqrt{108} \)
Prime factorize 12 and 108: \(12 = 2^2 \cdot 3\) and \(108 = 2^2 \cdot 3^3\).
\[ -4\sqrt{2^2 \cdot 3} + 12\sqrt{2^2 \cdot 3^2 \cdot 3} \] \[ = -8\sqrt{3} + 72\sqrt{3} = 64\sqrt{3} \]Example 7: \( \sqrt[4]{(x+1)} + 3\sqrt[4]{16(x+1)} \)
Practice Questions with Solutions
Question 1: \( -2\sqrt{3}+4\sqrt{3}+20 \)
Question 2: \( 20\sqrt{7}-2\sqrt{28}-7 \)
Question 3: \( -\sqrt{32}-2\sqrt{50}+3\sqrt{200} \)
Simplify: \(32 = 2 \cdot 16\), \(50 = 2 \cdot 25\), \(200 = 2 \cdot 100\).
\[ -4\sqrt{2} - 10\sqrt{2} + 30\sqrt{2} = 16\sqrt{2} \]