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Addition and Subtraction of Radicals

Add and subtract fractions with variables in the denominator

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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? 

Adding and Subtracting Radicals

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*}

Let's simplify the following expressions:

  1. \begin{align*}3\sqrt{5}+6\sqrt{5}\end{align*}.

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*}

  1. \begin{align*}2\sqrt[3]{13} + 6 \sqrt[3]{12}\end{align*}.

The cube roots are not like terms, so there can be no further simplification.

  1. \begin{align*}4\sqrt{3}+2\sqrt{12}\end{align*}.

In some cases, the radical may need to be reduced before addition/subtraction is possible.

\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*}


Example 1

Earlier, you were asked to find the total distance you travel in a trip. You know that the 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. 

To calculate the total distance you traveled, you need to add the radicals. 

\begin{align*}Total\ distance\ traveled=14\sqrt{2}+9\sqrt{2}\\ Total\ distance\ traveled=23\sqrt{2}\\\end{align*}

Example 2


\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*}


Write the following expressions in simplest radical form.

  1. \begin{align*}\sqrt[3]{48a^3b^7}\end{align*}
  2. \begin{align*}\sqrt[3]{\frac{16x^5}{135y^4}}\end{align*}
  3. 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.

  1. \begin{align*}3\sqrt{8}-6\sqrt{32}\end{align*}
  2. \begin{align*}\sqrt{180}+6\sqrt{405}\end{align*}
  3. \begin{align*}\sqrt{6}-\sqrt{27}+2\sqrt{54}+3\sqrt{48}\end{align*}
  4. \begin{align*}\sqrt{8x^3}-4x\sqrt{98x}\end{align*}
  5. \begin{align*}\sqrt{48a}+\sqrt{27a}\end{align*}
  6. \begin{align*}\sqrt[3]{4x^3}+x\sqrt[3]{256}\end{align*}

Mixed Review

  1. 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*}?
  2. Solve \begin{align*}\frac{x+3}{6}=\frac{21}{x}\end{align*}.
  3. 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?
  4. What is the vertex of \begin{align*}y=2(x+1)^2+4\end{align*}? Is this a minimum or a maximum?
  5. Using the minimum wage data (adjusted for inflation) compiled from EPI, answer the following questions.
    1. Graph the data as a scatter plot.
    2. Which is the best model for this data: linear, quadratic, or exponential?
    3. Find the model of best fit and use it to predict minimum wage adjusted for inflation for 1999.
    4. According to EPI, the 1999 minimum wage adjusted for inflation was $6.58. How close was your model?
    5. 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

Review (Answers)

To see the Review answers, open this PDF file and look for section 11.3. 

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For \sqrt{x}, x is the radicand. It is the value or variable inside the radical symbol.

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