Have you ever watched a baseball game? Have you ever seen a runner run so fast that he passed the intended base? Take a look at what Miguel saw.

Miguel was watching a game on Saturday night. During the fourth inning, the runner ran from second base to third base. He was running to fast that he ran past third base and had to come back to reach the base.

The runner ran \begin{align*}90 + \sqrt{16}\end{align*}

How far did the runner run?

**This situation has been described using a radical expression. You will learn how to evaluate radical expressions in this Concept.**

### Guidance

Sometimes, we can have an expression with a ** radical** in it.

**What is a radical?**

**A** *radical***is the name of the sign that tells us that we are looking for a square root.** We can call this “a radical.” Here is a radical symbol.

\begin{align*}\sqrt{y}\end{align*}

Here we would be looking for the square root of \begin{align*}y\end{align*}

**What is a radical expression?**

Remember than an expressions is a number sentence that contains numbers, operations and now radicals. Just as we can have expressions without radicals, we can have expressions with them too.

**Let's look at one.**

\begin{align*}2 \cdot \sqrt{4} + 7\end{align*}

Here we have two times the square root of four plus seven.

**That’s a great question! We can evaluate this expression by using the order of operations.**

**P parentheses**

**E exponents (square roots too)**

**MD multiplication/division in order from left to right**

**AS addition/subtraction in order from left to right**

According to the order of operations, we evaluate the square root of 4 first.

\begin{align*}\sqrt{4} = 2\end{align*}

Next, we substitute that value into the expression.

\begin{align*}2 \cdot 2 + 7\end{align*}

Next, we complete multiplication/division in order from left to right.

\begin{align*}2 \times 2 = 4\end{align*}

Substitute that given value.

\begin{align*}4 + 7 = 11\end{align*}

**The answer is 11.**

Let’s look at another one.

\begin{align*}\sqrt{4} \cdot \sqrt{16} - 3\end{align*}

Here we have two radicals in the expression. We can work the same way, by using the order of operations.

Evaluate the radicals first.

\begin{align*}\sqrt{4} & = 2\\ \sqrt{16} & = 4\end{align*}

Substitute these values into the expression.

\begin{align*}2 \cdot 4 - 3\end{align*}

Next, we complete multiplication/division in order from left to right.

\begin{align*}2 \times 4 = 8\end{align*}

Finally, we complete the addition/subtraction in order from left to right.

\begin{align*}8 - 3 = 5\end{align*}

**The answer is 5.**

Evaluate each radical expression.

#### Example A

\begin{align*}6 + \sqrt{9} - \sqrt{49} + 5\end{align*}

**Solution: \begin{align*}7\end{align*}**

#### Example B

\begin{align*}\sqrt{64} \div \sqrt{4} + 13\end{align*}

**Solution: \begin{align*}17\end{align*}**

#### Example C

\begin{align*}6(7) + \sqrt{121} - 3\end{align*}

**Solution: \begin{align*}50\end{align*}**

Here is the original problem once again.

Miguel was watching a game on Saturday night. During the fourth inning, the runner ran from second base to third base. He was running to fast that he ran past third base and had to come back to reach the base.

The runner ran \begin{align*}90 + \sqrt{16}\end{align*}.

How far did the runner run?

To figure this out, we have to evaluate the radical expression.

The distance from one base to another is 90 feet.

The extra distance is the \begin{align*}\sqrt{16}\end{align*}. Let's evaluate that part first.

\begin{align*}\sqrt{16} = 4\end{align*}

Now let's add that into the original expression.

\begin{align*}90 + 4 = 94\end{align*}

**The runner ran 94 feet.**

### Vocabulary

- Square Number
- a number of units which makes a perfect square.

- Square root
- a number that when multiplied by itself equals the square of the number.

- Perfect Square
- square roots that are whole numbers.

- Radical
- the symbol that lets us know that we are looking for a square root.

- Radical Expression
- an expression with numbers, operations and radicals in it.

### Guided Practice

Here is one for you to try on your own.

Evaluate this radical expression.

\begin{align*}8(7) + \sqrt{144} - 9\end{align*}

**Answer**

First, let's evaluate the square root.

\begin{align*}\sqrt{144} = 12\end{align*}

Now we can substitute 12 back into the original expression and solve using the order of operations.

\begin{align*}8(7) + 12 - 9\end{align*}

\begin{align*}56 + 12 - 9\end{align*}

\begin{align*}68 - 9 = 59\end{align*}

**Our answer is \begin{align*}59\end{align*}.**

### Video Review

- This is a Khan Academy video on radical expressions.

### Practice

Directions: Evaluate each radical expression.

1. \begin{align*}2 + \sqrt{9} + 15-2\end{align*}

2. \begin{align*}3 \cdot 4 + \sqrt{169}\end{align*}

3. \begin{align*}\sqrt{16} \cdot \sqrt{25} + \sqrt{36}\end{align*}

4. \begin{align*}\sqrt{81} \cdot 12 + 12\end{align*}

5. \begin{align*}\sqrt{36} + \sqrt{47} - \sqrt{16}\end{align*}

6. \begin{align*}6 + \sqrt{36} + 25-2\end{align*}

7. \begin{align*}4(5) + \sqrt{9}-2\end{align*}

8. \begin{align*}15 + \sqrt{16} + 5\end{align*}

9. \begin{align*}3(2) + \sqrt{25} + 10\end{align*}

10. \begin{align*}4(7) + \sqrt{49}-12\end{align*}

11. \begin{align*}2(4) + \sqrt{9}-8\end{align*}

12. \begin{align*}3(7) + \sqrt{25} + 21\end{align*}

13. \begin{align*}8(3) - \sqrt{36} + 15-2\end{align*}

14. \begin{align*}19 + \sqrt{144}-22\end{align*}

15. \begin{align*}3(4) + \sqrt{64} + \sqrt{25}\end{align*}