Algebra 1 - Adding and Subtracting Radical Expressions


  • Expressions that contain radicals are called radical expressions.
  • Recall that a radical has a radicand and an index apart from the radical symbol, as shown in the figure below.

  • Radicals are of two types - like radicals & unlike radicals.
    • Like radicals: Radicals with the same index and radicand are called like radicals (or similar radicals). For example 73 and 573 are a pair of like radicals.
    • Unlike radicals: Radicals with different indices and radicands are called unlike radicals (or dissimilar radicals). For example 2 and 53 are a pair of unlike radicals.

Addition and Subtraction of Radical Expressions

  • Only like radicals can be added or subtracted.
  • Follow the steps mentioned below to add or subtract radical expressions:
    • Simplify each radical term, if possible.
    • Combine the like terms.
  • Only the coefficients of like terms are added when adding terms with like radicals. The radical part remains unchanged.
  • Similar is the case with the subtraction of like radicals.

Some Solved Examples

Question 1: Perform the indicated operation: 5xy+7xy-2xy

Solution: 5xy+7xy-2xy=12xy-2xy=10xy

Question 2: Perform the indicated operation: 4 54x43-5x 250x3-3x 128x3

Solution: 4 54x43-5x 250x3-3x 128x3=4 27·2x43-5x 125·2x3-3x 64·2x3=12x 2x3-25x 2x3-12x 2x2=-25x 2x3

Question 3: Perform the indicated operation: 6yx+4yx-2 x3+4 x3

Solution: 6yx+4yx-2 x3+4 x3=10yx+2 x3

Question 4: Simplify the expression: 412xy4-y147xy2

Solution: 412xy4-y147xy2=44·3xy4-y49·3xy2=8y2 3x-7y2 3x=y2 3x

Question 5: Simplify the expression: 8x-83+27x-273-64x+643

Solution: 8x-83+27x-273-64x+643=8x-13+27x-13-64x+13=2 x-13+3 x-13-4 x+13=5 x-13-4 x+13

Question 6: Simplify the expression: n6-n4-n24n2-4

Solution: n6-n4-n24n2-4=n4n2-1-n24n2-1=n2n2-1-2n2n2-1=-n2n2-1

Question 7: What is the simplified form of 47x2y-5x2y3+63x5?

Solution: 47x2y-5x2y3+63x5=4x7y-5xyy+3x27x

Cheat Sheet

  • Only like radicals can be added or subtracted.
  • Generally, we will have to simplify each term of the radical expression before we can identify the like terms.
  • In addition or subtraction of radical expression, we only add or subtract the coefficients of the like terms, and the radicand part remains as it is.

Blunder Areas

  • 57+671114 because only the coefficients of like terms are added & the radicands remain the same. 
  • Likewise, 813x-411x42x.
  • When simplifying and performing operations in radicals and the expression containing polynomial radicands, we always assume that the variables involved are greater than 0.
  • Don't assume that expressions with unlike radicals cannot be simplified. It is possible that, after simplifying the radicals, the expression can indeed be simplified.

Enrichment: Nested Radicals

  • A nested radical is a radical expression that contains another inner radical expression.
  • Nested Radicals can be Finite or Infinite.
  • Some examples of finite nested radicals are a-b, 4+3, 2+35, x434.
  • Some examples of infinite nested radicals are 3+3+3+3+..., 5-5-5-5...
  • The goal of simplifying these types of expression is to denest the radical. 
  • To denest the form a±b, the following conditions must be satisfied:
    • a and b are positive and rational.
    • a2-b is a perfect square.
    • The inner radical b must be irrational.
    • If a-b, then a2b. For a>0 and a2b, the following formula will be used: a+b=a+a2-b2+a-a2-b2


  • Shortcut formulas for the case of a+2b, we have: a+2b=a+b+2ab=a+b where a>0 and b>0.
  • One common formula for simplifying infinite nested radical is x+x+x+x+...=121+4x+1, where x>0