Mark, a huge baseball fan, asked his sister Sara, “Did you know that the distance between any two bases on a baseball diamond is equal to the square root of 8100?”

How can Sara learn or estimate the square root of 8100?

In this concept, you will learn to evaluate radical expressions.

### Radical Expressions

An **exponent** is the small number to the upper right of a base that tells you how many times to multiply the base times itself. For example \begin{align*}7^2 = 7 \times 7 = 49\end{align*}. The base of 7 is raised to the power of 2 which means to multiply 7 times itself twice. The answer is 49.

Finding the square root of a number is the inverse operation of squaring. The **square root** of a number ‘ ,’ denoted by \begin{align*}\sqrt{n}\end{align*}, is a positive number whose square is \begin{align*}n\end{align*}. The \begin{align*}\sqrt{n}\end{align*} is called a **radical.** A **radical** is an expression consisting of a **radical sign** \begin{align*}\sqrt{}\end{align*} , a **radicand** which is the number or term under the radical sign and an **index** which is the small number outside the radical sign which indicates what root to find. The index for ‘square root’ is two but it is never written in the expression - it is understood since the radical sign with no index is referred to as the square root.

The following is an example of a radical.

\begin{align*}\begin{array}{rcl} && \qquad \ \ \ \ \ \downarrow \text{radical sign}\\ && {\color{red}\text{index} \rightarrow} \sqrt[{\color{red}3}]{{\color{blue}64}} {\color{blue}\leftarrow \text{radicand}} \end{array}\end{align*}

When a number is written under a radical sign, you are finding the square root of that number. For the radical **perfect square** because it is a number that can be written as a power with an integer base and the exponent 2.

A number raised to the power of 3 is said to be **cubed**. For example \begin{align*} 4^3 = 4 \times 4 \times 4 = 64\end{align*}. The base of 4 has been multiplied by itself three times. The inverse operation would involve finding the cube root of 64

The **cube root** of 64 is the number times itself three times that equals 64. The expression can be written as \begin{align*} \sqrt[3]{64} =3\end{align*} . Sixty-four is called a **perfect cube** because it is a number that can be written as a power with an integer base and the exponent 3.

Not all numbers are perfect squares or perfect cubes. The values for these numbers can either be estimated or found using the TI calculator. If you were asked to find the square root of thirty then you could estimate the answer.

\begin{align*}\sqrt{30}\end{align*} is between the two perfect squares . Since 30 is approximately half-way between 25 and 36, then an estimate of 5.5 would be a reasonable answer for the square root of 30.

The estimate of 5.5 is the same as the calculator answer if it were rounded to the nearest tenth.

Estimate the following:

\begin{align*}\sqrt[3]{35}\end{align*}

Thirty-five is not a perfect cube since there is no integer when multiplied by itself 3 times equal to 35.

The value 35 is between the perfect cube \begin{align*}27= 3 \times 3 \times 3 = 3 ^3 \end{align*} and \begin{align*}64= 4 \times 4 \times 4 = 4 ^ 3\end{align*}. The number 35 is closer to 27 than it is to 64. Therefore an estimate for the cube root of 35 is 3.3.

### Examples

#### Example 1

Earlier, you were given a problem about Mark and the baseball diamond. His sister wants to figure out the square root of 8100.

First, rewrite the square root of 8100 as a radical.

\begin{align*}\sqrt{8100}\end{align*}

Next, write the square root of 8100 as \begin{align*}\sqrt{81} \times \sqrt{100} \end{align*}

Next, write the square of 81.

\begin{align*}81= 9 \times 9 = 9^2 \end{align*} The square of 81 is 9.

Next, write the square of 100.

\begin{align*}100 = 10 \times 10 = 10^2\end{align*} The square of 100 is 10.

Then, multiply the two squares.

\begin{align*}9 \times 10 = 90\end{align*}

The answer is 90.

The distance between any two bases is 90 feet.

#### Example 2

Estimate the square root of the following by listing the two squares that the root falls between:

\begin{align*}\sqrt{74}\end{align*}

First, figure out the perfect square that is closest to but less than 74.

\begin{align*}64 = 8 ^2\end{align*}

Next, figure out the perfect square that is closest to but greater than 74.

\begin{align*}81 = 9^2\end{align*}

The answer is 8 and 9.

\begin{align*}\sqrt{74}\end{align*} is between 8 and 9.

#### Example 3

Using the TI calculator, calculate the following square root to the nearest hundredth.

\begin{align*}\sqrt{115}\end{align*}

First, press the buttons \begin{align*} 2 ^{nd} \ x^2 \end{align*} 1 1 5 **enter **

Next, look at the answer displayed on the screen of the calculator.

\begin{align*}10.72380529\end{align*}

Then, round the answer to two places after the decimal point.

\begin{align*}10.73\end{align*}

The answer is \begin{align*} \sqrt{115} \approx 10.73\end{align*}

#### Example 4

Without using technology, which of the following numbers are perfect squares? Justify your answer.

\begin{align*}196 \ \ \ 120 \ \ \ 36\end{align*}

Create a factor tree for each of the given numbers.

First, 196 is an even number so divide it by two.

\begin{align*}\frac{196}{2} = 98 \end{align*}

Next, 98 is an even number so divide it by two.

\begin{align*}\frac{98}{2} = 49 \end{align*}

Next, write down the numbers that will multiply to give 49.

\begin{align*}7 \times 7\end{align*}

Next, list all the factors used to simplify 196.

\begin{align*}2,2,7,7\end{align*}

Next, make two groups of same factors.

\begin{align*}2 \times 7 \text{ and }2 \times 7\end{align*}

Then, multiply each group of factors.

\begin{align*}14 \ \text{and} \ 14\end{align*}

The answer is 14.

\begin{align*} \sqrt{196} = 14\end{align*}

First, 120 is an even number so divide it by two.

\begin{align*}\frac{120}{2} = 60\end{align*}

Next, 60 is an even number so divide it by two.

\begin{align*}\frac{60}{2} = 30\end{align*}

Next, 30 is an even number so divide it by two.

\begin{align*}\frac{30}{2} = 15\end{align*}

Next, write down the numbers that multiply to give 15.

\begin{align*}3 \times 5\end{align*}

Next, list all the factors used to simplify 120.

\begin{align*}2, 2, 2, 3, 5\end{align*}

Then, make two groups of same factors.

There are three 2’s, one 3, and one 5. Two groups of same factors cannot be made from these numbers.

120 is not a perfect square.

First, 36 is an even number so divide it by two.

\begin{align*}\frac{36}{2} = 18\end{align*}

Next, 18 is an even number so divide it by two.

\begin{align*}\frac{18}{2} = 9\end{align*}

Next, write down the numbers that multiply to give 9.

\begin{align*}3 \times 3\end{align*}

Next, list all the factors used to simplify 36.

\begin{align*}2, 2, 3, 3\end{align*}

Next, make two groups of same factors.

\begin{align*}2 \times 3 \text{ and } 2 \times 3\end{align*}

Then, multiply each group of factors.

\begin{align*}6 \ \text{and} \ 6\end{align*}

The answer is 6.

\begin{align*}\sqrt{36} = 6\end{align*}

### Review

Evaluate each radical expression without using technology.

1.

2. \begin{align*}\sqrt{25}\end{align*}

3.

4.

5.

6.

7.

8.

9.

10.

Approximate each square root by listing the two values that the square root can be found between.

11.

12.

13.

14.

15. \begin{align*}\sqrt{31}\end{align*}

16.

17.