10.2: Add and subtract radicals
- Page ID
- 45135
Adding and subtracting radicals are very similar to adding and subtracting with variables. In order to combine terms, they need to be like terms. With radicals, we have something similar called like radicals. Let’s look at an example with like terms and like radicals.
\[\begin{array}{cc}2\color{blue}{x}\color{black}{+}5\color{blue}{x}&\color{black}{2}\color{blue}{\sqrt{3}}\color{black}{+}5\color{blue}{\sqrt{3}} \\ (2+5)\color{blue}{x}&\color{black}{(}2+5)\color{blue}{\sqrt{3}} \\ 7\color{blue}{x}&\color{black}{7}\color{blue}{\sqrt{3}}\end{array}\nonumber\]
Notice that when we combined the terms with \(\sqrt{3}\), it was similar to combining terms with \(x\). When adding and subtracting with radicals, we can combine like radicals just as like terms.
If two radicals have the same radicand and the same root, then they are called like radicals. If this is so, then \[a\sqrt{x}\pm b\sqrt{x}=(a\pm b)\sqrt{x},\nonumber\] where \(a\), \(b\) are real numbers and \(x\) is some positive real number.
In general, for any root \(n\), \[a\sqrt[n]{x}\pm b\sqrt[n]{x}=(a\pm b)\sqrt[n]{x},\nonumber\] where \(a\), \(b\) are real numbers and \(x\) is some positive real number.
When simplifying radicals with addition and subtraction, we will simplify the expression first, then extract out any factors from the radicand following the guidelines in the previous section.
Add and Subtract like Radicals
Simplify: \(7\sqrt[5]{6}+4\sqrt[5]{3}-9\sqrt[5]{3}+\sqrt[5]{6}\)
Solution
Notice, all the indices are the same, but two of the radicands are different. We only combine like radicals, where the root and radicand are the same.
\[\begin{array}{rl}7\sqrt[5]{6}+4\sqrt[5]{3}-9\sqrt[5]{3}+\sqrt[5]{6}&\text{Combine the like radicals} \\ (7+1)\color{blue}{\sqrt[5]{6}}\color{black}{+}(4-9)\color{blue}{\sqrt[5]{3}}&\color{black}{\text{Simplify}} \\ 8\color{blue}{\sqrt[5]{6}}\color{black}{-}5\color{blue}{\sqrt[5]{3}}&\color{black}{\text{Simplified expression}}\end{array}\nonumber\]
Notice, radicands \(6\) and \(3\) have no factors that are perfect \(5^{\text{th}}\) powers. Thus, the expression is completely simplified.
Simplify, then Add and Subtract like Radicals
Simplify: \(5\sqrt{45}+6\sqrt{18}-2\sqrt{98}+\sqrt{20}\)
Solution
Notice, all the indices are the same, but none of the radicands are the same. However, we can see that the radicands have factors that are perfect squares. We can simplify the radicands first, then see if we can combine like radicals.
\[\begin{array}{rl}5\sqrt{45}+6\sqrt{18}-2\sqrt{98}+\sqrt{20}&\text{Rewrite radicand} \\ 5\cdot\sqrt{9\cdot 5}+6\cdot\sqrt{9\cdot 2}-2\cdot\sqrt{49\cdot 2}+\sqrt{4\cdot 5}&\text{Apply product rule for radicals} \\ 5\cdot\sqrt{9}\cdot\sqrt{5}+6\cdot\sqrt{9}\cdot\sqrt{2}-2\cdot\sqrt{49}\cdot\sqrt{2}+\sqrt{4}\cdot\sqrt{5}&\text{Simplify each square root} \\ 5\cdot 3\cdot\sqrt{5}+6\cdot 3\cdot\sqrt{2}-2\cdot 7\cdot\sqrt{2}+2\cdot\sqrt{5}&\text{Rewrite and simplify coefficients} \\ 15\sqrt{5}+18\sqrt{2}-14\sqrt{2}+2\sqrt{5}&\text{Combine the like radicals} \\ (15+2)\color{blue}{\sqrt{5}}\color{black}{+}(18-14)\color{blue}{\sqrt{2}}&\color{black}{\text{Simplify}} \\ 17\color{blue}{\sqrt{5}}\color{black}{+}4\color{blue}{\sqrt{2}}&\color{black}{\text{Simplified expression}}\end{array}\nonumber\]
The Arab writers of the \(16^{\text{th}}\) century used the symbol similar to the greater than symbol with a dot underneath, \(\stackrel{<}{·}\), for radicals.
Simplify: \(4\sqrt[3]{54}-9\sqrt[3]{16}+5\sqrt[3]{9}\)
Solution
We apply the same method as the previous examples, but the root is \(3\) and we will look for the largest factor of the radicand that is a perfect cube when simplifying the radicals.
\[\begin{array}{rl}4\sqrt[3]{54}-9\sqrt[3]{16}+5\sqrt[3]{9}&\text{Rewrite radicand} \\ 4\cdot\sqrt[3]{27\cdot 2}-9\cdot\sqrt[3]{8\cdot 2}+5\cdot\sqrt[3]{9}&\text{Apply product rule for radicals and simplify} \\ 4\cdot 3\sqrt[3]{2}-9\cdot 2\sqrt[3]{2}+5\sqrt[3]{9}&\text{Rewrite and simplify coefficients} \\ 12\sqrt[3]{2}-18\sqrt[3]{2}+5\sqrt[3]{9}&\text{Combine the like radicals} \\ (12-18)\color{blue}{\sqrt[3]{2}}\color{black}{+}5\sqrt[3]{9}&\text{Simplify} \\ -6\sqrt[3]{2}+5\sqrt[3]{9}&\text{Simplified expression}\end{array}\nonumber\]
Add and Subtract Radicals Homework
Simplify.
\(2\sqrt{5}+2\sqrt{5}+2\sqrt{5}\)
\(-3\sqrt{2}+3\sqrt{5}+3\sqrt{5}\)
\(-2\sqrt{6}-2\sqrt{6}-\sqrt{6}\)
\(3\sqrt{6}+3\sqrt{5}+2\sqrt{5}\)
\(2\sqrt{2}-3\sqrt{18}-\sqrt{2}\)
\(-3\sqrt{6}-\sqrt{12}+3\sqrt{3}\)
\(3\sqrt{2}+2\sqrt{8}-3\sqrt{18}\)
\(3\sqrt{18}-\sqrt{2}-3\sqrt{2}\)
\(-3\sqrt{6}-3\sqrt{6}-\sqrt{3}+3\sqrt{6}\)
\(-2\sqrt{18}-3\sqrt{8}-\sqrt{20}+2\sqrt{20}\)
\(-2\sqrt{24}-2\sqrt{6}+2\sqrt{6}+2\sqrt{20}\)
\(3\sqrt{24}-3\sqrt{27}+2\sqrt{6}+2\sqrt{8}\)
\(-2\sqrt[3]{16}+2\sqrt[3]{16}+2\sqrt[3]{2}\)
\(2\sqrt[4]{243}-2\sqrt[4]{243}-\sqrt[4]{3}\)
\(3\sqrt[4]{2}-2\sqrt[4]{2}-\sqrt[4]{243}\)
\(-\sqrt[4]{324}+3\sqrt[4]{324}-3\sqrt[4]{4}\)
\(2\sqrt[4]{2}+2\sqrt[4]{3}+3\sqrt[4]{64}-\sqrt[4]{3}\)
\(-3\sqrt[5]{6}-\sqrt[5]{64}+2\sqrt[5]{192}-2\sqrt[5]{64}\)
\(2\sqrt[5]{160}-2\sqrt[5]{192}-\sqrt[5]{160}-\sqrt[5]{-160}\)
\(-\sqrt[6]{256}-2\sqrt[6]{4}-3\sqrt[6]{320}-2\sqrt[6]{128}\)
\(-3\sqrt{6}-3\sqrt{3}-2\sqrt{3}\)
\(-2\sqrt{6}-\sqrt{3}-3\sqrt{6}\)
\(-3\sqrt{3}+2\sqrt{3}-2\sqrt{3}\)
\(-\sqrt{5}+2\sqrt{3}-2\sqrt{3}\)
\(-\sqrt{54}-3\sqrt{6}+3\sqrt{27}\)
\(-\sqrt{5}-\sqrt{5}-2\sqrt{54}\)
\(2\sqrt{20}+2\sqrt{20}-\sqrt{3}\)
\(-3\sqrt{27}+2\sqrt{3}-\sqrt{12}\)
\(-2\sqrt{2}-\sqrt{2}+3\sqrt{8}+3\sqrt{6}\)
\(-3\sqrt{18}-\sqrt{8}+2\sqrt{8}+2\sqrt{8}\)
\(-3\sqrt{8}-\sqrt{5}-3\sqrt{6}+2\sqrt{18}\)
\(2\sqrt{6}-\sqrt{54}-3\sqrt{27}-\sqrt{3}\)
\(3\sqrt[3]{125}-\sqrt[3]{81}-\sqrt[3]{135}\)
\(-3\sqrt[4]{4}+3\sqrt[4]{324}+2\sqrt[4]{64}\)
\(2\sqrt[4]{6}+2\sqrt[4]{4}+3\sqrt[4]{6}\)
\(-2\sqrt[4]{243}-\sqrt[4]{96}+2\sqrt[4]{96}\)
\(2\sqrt[4]{48}-3\sqrt[4]{405}-3\sqrt[4]{48}-\sqrt[4]{162}\)
\(-3\sqrt[7]{3}-3\sqrt[7]{768}+2\sqrt[7]{384}+3\sqrt[7]{5}\)
\(-2\sqrt[7]{256}-2\sqrt[7]{256}-3\sqrt[7]{2}-\sqrt[7]{640}\)