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:

- \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*}

- \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.

- \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*}

### Examples

#### 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*}3\sqrt[3]{2}+5\sqrt[3]{16}\end{align*}.

\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*}

### Review

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 |

### Review (Answers)

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