11.3: Addition and Subtraction of Radicals
Suppose that you're taking a trip, and you'll be making two stops. This distance from your starting point to your first stop is \begin{align*}14 \sqrt{2}\end{align*} miles, and the distance from your first stop to your second stop is \begin{align*}9 \sqrt{2}\end{align*} miles. How far will you travel in total? What operation would you have to perform to find the answer to this question? In this Concept, you'll learn how to add and subtract radicals like the ones given here.
Guidance
To add or subtract radicals, they must have the same root and radicand.
\begin{align*}a \sqrt[n]{x}+b\sqrt[n]{x}=(a+b)\sqrt[n]{x}\end{align*}
Example A
Add: \begin{align*}3\sqrt{5}+6\sqrt{5}\end{align*}.
Solution:
The value “\begin{align*}\sqrt{5}\end{align*}” is considered a like term. Using the rule above:
\begin{align*}3 \sqrt{5}+6\sqrt{5}=(3+6) \sqrt{5}=9\sqrt{5}\end{align*}
Example B
Simplify \begin{align*}2\sqrt[3]{13} + 6 \sqrt[3]{12}\end{align*}.
Solution:
The cube roots are not like terms, so there can be no further simplification.
In some cases, the radical may need to be reduced before addition/subtraction is possible.
Example C
Simplify \begin{align*}4\sqrt{3}+2\sqrt{12}\end{align*}.
Solution:
\begin{align*}\sqrt{12}\end{align*} simplifies to \begin{align*}2\sqrt{3}\end{align*}.
\begin{align*}4\sqrt{3}+2\sqrt{12} &\rightarrow 4\sqrt{3}+2\left ( 2\sqrt{3} \right )\\ 4\sqrt{3}+4\sqrt{3}&=8\sqrt{3}\end{align*}
Guided Practice
Add: \begin{align*}3\sqrt[3]{2}+5\sqrt[3]{16}\end{align*}.
Solution:
\begin{align*}\text{Begin by factoring the second radical.} && 3\sqrt[3]{2}+5\sqrt[3]{16}&=3\sqrt[3]{2}+5\sqrt[3]{2\cdot 8}=3\sqrt[3]{2}+5\sqrt[3]{2\cdot 2^3}\\ \text{Simplify the second radical using properties of roots.} && &=3\sqrt[3]{2}+5 \sqrt[3]{2^3}\cdot\sqrt[3]{2}=3\sqrt[3]{2}+5\cdot 2\sqrt[3]{2} =3\sqrt[3]{2}+10\sqrt[3]{2}\\ \text{The terms are now alike and can be added.} && &=(3+10)\sqrt[3]{2}=13 \sqrt[3]{2} \end{align*}
Practice
Sample explanations for some of the practice exercises below are available by viewing the following videos. Note that there is not always a match between the number of the practice exercise in the videos and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both.
CK-12 Basic Algebra: More Simplifying Radical Expressions (7:57)
Write the following expressions in simplest radical form.
- \begin{align*}\sqrt[3]{48a^3b^7}\end{align*}
- \begin{align*}\sqrt[3]{\frac{16x^5}{135y^4}}\end{align*}
- True or false? \begin{align*}\sqrt[7]{5} \cdot \sqrt[6]{6}=\sqrt[42]{30}\end{align*}
Simplify the following expressions as much as possible.
- \begin{align*}3\sqrt{8}-6\sqrt{32}\end{align*}
- \begin{align*}\sqrt{180}+6\sqrt{405}\end{align*}
- \begin{align*}\sqrt{6}-\sqrt{27}+2\sqrt{54}+3\sqrt{48}\end{align*}
- \begin{align*}\sqrt{8x^3}-4x\sqrt{98x}\end{align*}
- \begin{align*}\sqrt{48a}+\sqrt{27a}\end{align*}
- \begin{align*}\sqrt[3]{4x^3}+x\sqrt[3]{256}\end{align*}
Mixed Review
- An item originally priced \begin{align*}\$c\end{align*} is marked down 15%. The new price is $612.99. What is \begin{align*}c\end{align*}?
- Solve \begin{align*}\frac{x+3}{6}=\frac{21}{x}\end{align*}.
- According to the Economic Policy Institute (EPI), minimum wage in 1989 was $3.35 per hour. In 2009, it was $7.25 per hour. What is the average rate of change?
- What is the vertex of \begin{align*}y=2(x+1)^2+4\end{align*}? Is this a minimum or a maximum?
- Using the minimum wage data (adjusted for inflation) compiled from EPI, answer the following questions.
- Graph the data as a scatter plot.
- Which is the best model for this data: linear, quadratic, or exponential?
- Find the model of best fit and use it to predict minimum wage adjusted for inflation for 1999.
- According to EPI, the 1999 minimum wage adjusted for inflation was $6.58. How close was your model?
- Use interpolation to find the minimum wage in 1962.
Year | Minimum Wage Adj. for Inflation | Year | Minimum Wage Adj. for Inflation |
---|---|---|---|
1947 | 3.40 | 1952 | 5.36 |
1957 | 6.74 | 1960 | 6.40 |
1965 | 7.52 | 1970 | 7.81 |
1978 | 7.93 | 1981 | 7.52 |
1986 | 6.21 | 1990 | 6.00 |
1993 | 6.16 | 1997 | 6.81 |
2000 | 6.37 | 2004 | 5.80 |
2006 | 5.44 | 2008 | 6.48 |
2009 | 7.25 |
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Image Attributions
Here you'll learn how to add and subtract radicals in order to simplify expressions.