Algebra 1 - Adding and Subtracting Radical Expressions

Introduction

• 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 $\sqrt[3]{7}$ and $5\sqrt[3]{7}$ are a pair of like radicals.
  • Unlike radicals: Radicals with different indices and radicands are called unlike radicals (or dissimilar radicals). For example $\sqrt{2}$ and $\sqrt[3]{5}$ 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: $5x\sqrt{y}+7x\sqrt{y}-2x\sqrt{y}$

Solution: $5x\sqrt{y}+7x\sqrt{y}-2x\sqrt{y}=12x\sqrt{y}-2x\sqrt{y}=\mathbf{10}\mathit{x}\sqrt{\mathbf{y}}$

Question 2: Perform the indicated operation: $4 \sqrt[3]{54x^4}-5x \sqrt[3]{250x}-3x \sqrt[3]{128x}$

Solution: $4 \sqrt[3]{54x^4}-5x \sqrt[3]{250x}-3x \sqrt[3]{128x}$$=4 \sqrt[3]{27·2x^4}-5x \sqrt[3]{125·2x}-3x \sqrt[3]{64·2x}$$=12x \sqrt[3]{2x}-25x \sqrt[3]{2x}-12x \sqrt[2]{2x}$$=\mathbf{-}\mathbf{25}\mathit{x}\mathbf{ }\sqrt[\mathbf{3}]{\mathbf{2}\mathbf{x}}$

Question 3: Perform the indicated operation: $6y\sqrt{x}+4y\sqrt{x}-2 \sqrt[3]{x}+4 \sqrt[3]{x}$

Solution: $6y\sqrt{x}+4y\sqrt{x}-2 \sqrt[3]{x}+4 \sqrt[3]{x}$$=\mathbf{10}\mathit{y}\sqrt{\mathbf{x}}\mathbf{+}\mathbf{2}\mathbf{ }\sqrt[\mathbf{3}]{\mathbf{x}}$

Question 4: Simplify the expression: $4\sqrt{12x{y}^4}-y\sqrt{147x{y}^2}$

Solution: $4\sqrt{12x{y}^4}-y\sqrt{147x{y}^2}$$=4\sqrt{4·3x{y}^4}-y\sqrt{49·3x{y}^2}$$=8y^2 \sqrt{3x}-7y^2 \sqrt{3x}$$={\mathit{y}}^{\mathbf{2}}\mathbf{ }\sqrt{\mathbf{3}\mathbf{x}}$

Question 5: Simplify the expression: $\sqrt[3]{8x-8}+\sqrt[3]{27x-27}-\sqrt[3]{64x+64}$

Solution: $\sqrt[3]{8x-8}+\sqrt[3]{27x-27}-\sqrt[3]{64x+64}$$=\sqrt[3]{8\left(x-1\right)}+\sqrt[3]{27\left(x-1\right)}-\sqrt[3]{64\left(x+1\right)}$$=2 \sqrt[3]{x-1}+3 \sqrt[3]{x-1}-4 \sqrt[3]{x+1}$$=\mathbf{5}\mathbf{ }\sqrt[\mathbf{3}]{\mathbf{x}\mathbf{-}\mathbf{1}}\mathbf{-}\mathbf{4}\mathbf{ }\sqrt[\mathbf{3}]{\mathbf{x}\mathbf{+}\mathbf{1}}$

Question 6: Simplify the expression: $\sqrt{n^6-n^4}-n^2\sqrt{4n^2-4}$

Solution: $\sqrt{n^6-n^4}-n^2\sqrt{4n^2-4}$$=\sqrt{n^4\left(n^2-1\right)}-n^2\sqrt{4\left(n^2-1\right)}$$=n^2\sqrt{n^2-1}-2n^2\sqrt{n^2-1}$$=\mathbf{-}{\mathit{n}}^{\mathbf{2}}\sqrt{{\mathbf{n}}^{\mathbf{2}}\mathbf{-}\mathbf{1}}$

Question 7: What is the simplified form of $4\sqrt{7x^{2}y}-5\sqrt{x^2{y}^3}+\sqrt{63x^5}$?

Solution: $4\sqrt{7x^{2}y}-5\sqrt{x^2{y}^3}+\sqrt{63x^5}$$=\mathbf{4}\mathit{x}\sqrt{\mathbf{7}\mathbf{y}}\mathbf{-}\mathbf{5}\mathit{x}\mathit{y}\sqrt{\mathbf{y}}\mathbf{+}\mathbf{3}{\mathit{x}}^{\mathbf{2}}\sqrt{\mathbf{7}\mathbf{x}}$

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

• $5\sqrt{7}+6\sqrt{7}\ne 11\sqrt{14}$ because only the coefficients of like terms are added & the radicands remain the same. 
• Likewise, $8\sqrt{13x}-4\sqrt{11x}\ne 4\sqrt{2x}$.
• 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 $\sqrt{a-\sqrt{b}}, \sqrt{4+\sqrt{3}}, \sqrt{2+3\sqrt{5}}, \sqrt[4]{\sqrt[3]{\sqrt{\sqrt[4]{x}}}}$.
• Some examples of infinite nested radicals are $\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+...}}}}, \sqrt{5-\sqrt{5-\sqrt{5-\sqrt{5...}}}}$
• The goal of simplifying these types of expression is to denest the radical. 
• To denest the form $\sqrt{a\pm \sqrt{b}}$, the following conditions must be satisfied:
  • $a and b$ are positive and rational.
  • $a^2-b$ is a perfect square.
  • The inner radical $\sqrt{b}$ must be irrational.
  • If $\sqrt{a-\sqrt{b}}, then a^2\ge b$. For $a>0 and a^2\ge b$, the following formula will be used: $\sqrt{a+\sqrt{b}}=\sqrt{\frac{a+\sqrt{a^2-b}}{2}}+\sqrt{\frac{a-\sqrt{a^2-b}}{2}}$

 

• Shortcut formulas for the case of $\sqrt{a+2\sqrt{b}}$, we have: $\sqrt{a+2\sqrt{b}}=\sqrt{a+b+2\sqrt{ab}}=\sqrt{a}+\sqrt{b} where a>0 and b>0$.
• One common formula for simplifying infinite nested radical is $\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+...}}}}=\frac{1}{2}\left(1+\sqrt{4x+1}\right), where x>0$