### Square Roots and Irrational Numbers

The **square root** of a number is a number which, when multiplied by itself, gives the original number. In other words, if

**Note:** Negative numbers and positive numbers both yield positive numbers when squared, so each positive number has both a positive and a negative square root. (For example, 3 and -3 can both be squared to yield 9.) The positive square root of a number is called the **principal square root**.

The square root of a number **radical sign**.

Numbers with whole-number square roots are called **perfect squares**. The first five perfect squares (1, 4, 9, 16, and 25) are shown below.

You can determine whether a number is a perfect square by looking at its prime factors. If every number in the factor tree appears an even number of times, the number is a perfect square. To find the square root of that number, simply take one of each pair of matching factors and multiply them together.

#### Finding the Principal Square Root

Find the principal square root of each of these perfect squares.

a) 121

b) 225

c) 324

When the prime factors don’t pair up neatly, we “factor out” the ones that do pair up and leave the rest under a radical sign. We write the answer as

#### Finding the Principal Square Roots of Imperfect Squares

Find the principal square root of the following numbers.

a) 8

b) 48

c) 75

Note that in the last example we collected the paired factors first, **then** we collected the unpaired ones under a single radical symbol. Here are the four rules that govern how we treat square roots.

a√×b√=ab−−√ Aa√×Bb√=ABab−−√ a√b√=ab−−√ Aa√Bb√=ABab−−√

#### Simplifying Square Roots

Simplify the following square root problems

a)

b)

c)

d)

**Approximate Square Roots**

Terms like **rational numbers**. That is to say, they cannot be expressed as the ratio of two integers. We call them **irrational numbers**. In decimal form, they have an unending, seemingly random, string of numbers after the decimal point.

To find approximate values for square roots, we use the **approximate answer**—not exactly the right answer, but as close as it can get.

#### Using a Calculator

Use a calculator to find the following square roots. Round your answer to three decimal places.

a)

b)

c)

d)

### Examples

Find the square root of each number.

#### Example 1

576

#### Example 2

216

### Review

For 1-10, find the following square roots **exactly without using a calculator**, giving your answer in the simplest form.

- \begin{align*}\sqrt{25}\end{align*}
- \begin{align*}\sqrt{24}\end{align*}
- \begin{align*}\sqrt{20}\end{align*}
- \begin{align*}\sqrt{200}\end{align*}
- \begin{align*}\sqrt{2000}\end{align*}
- \begin{align*}\sqrt{\frac{1}{4}}\end{align*} (Hint: The division rules you learned can be applied backwards!)
- \begin{align*}\sqrt{\frac{9}{4}}\end{align*}
- \begin{align*}\sqrt{0.16}\end{align*}
- \begin{align*}\sqrt{0.1}\end{align*}
- \begin{align*}\sqrt{0.01}\end{align*}

For 11-20, use a calculator to find the following square roots. Round to two decimal places.

- \begin{align*}\sqrt{13}\end{align*}
- \begin{align*}\sqrt{99}\end{align*}
- \begin{align*}\sqrt{123}\end{align*}
- \begin{align*}\sqrt{2}\end{align*}
- \begin{align*}\sqrt{2000}\end{align*}
- \begin{align*}\sqrt{.25}\end{align*}
- \begin{align*}\sqrt{1.35}\end{align*}
- \begin{align*}\sqrt{0.37}\end{align*}
- \begin{align*}\sqrt{0.7}\end{align*}
- \begin{align*}\sqrt{0.01}\end{align*}

### Review (Answers)

To view the Review answers, open this PDF file and look for section 2.9.