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ELM: CSU Math Study Guide17 chapters | 147 lessons | 7 flashcard sets

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Lesson Transcript

Instructor:
*Jennifer Beddoe*

Numbers that are imperfect squares are those that, when evaluated, do not give solutions that are integers. The proper mathematical way to simplify these imperfect squares is discussed in this lesson.

The study of mathematics is universal. It is studied the same whether you are in America, Russia, India or any other corner of the globe. To make it easy for all these mathematicians who speak different languages and come from different backgrounds, there are universal rules that apply to writing mathematical equations and the way in which all math is formatted. This way, mathematicians in Korea can understand the work that is being done in Canada.

One of these rules relates to how the square roots of imperfect squares are written. The proper mathematical way to write imperfect square roots is by simplifying them as much as possible without using a fraction or a decimal. This will result in numbers like the following:

4âˆš2

2âˆš7

Before we can work out how to simplify imperfect square roots, we need to do a bit of review about perfect and imperfect square roots.

A **perfect square** is a number whose square root is an integer. An **integer** is a number that does not contain a fraction or a decimal. The table below shows the first 10 perfect squares.

As you can see, only a few numbers are perfect squares. The rest can be classified as imperfect squares. **Imperfect squares** are numbers whose square roots contain fractions or decimals.

For example:

âˆš20 = 4 ½

âˆš77 = 8.77

A calculator will not simplify the square root of an imperfect square properly. It will give a result, but it will be in decimal form. So, to evaluate and simplify an imperfect square, you need to follow these steps:

1. Factor the number completely.

An easy way to factor a number is by using a **factor tree**. A factor tree can be created by writing down the number you want to factor and drawing two lines coming down from that number. Then, write two factors of that number under the lines. Continue on until only prime numbers remain. A **prime number** is one which cannot be reduced any smaller. The purpose of the factor tree is to determine which numbers can be removed from under the square root symbol.

2. Match up pairs of the same number.

Any numbers with a partner are perfect squares, and you can take the square root of these numbers.

3. Numbers without a partner remain under the square root.

These numbers cannot be simplified further.

For example, let's simplify the square root of 48. First, we factor 48. Then, we match up the pairs. As you can see below, there are two pairs of 2s. We can simplify both of these as 2 * 2 * 2 * 2, which is 16, and the square root of 16 equals 4. This means that there will be a 4 outside the square root symbol when we're done. The 3 does not have a partner, so it will remain under the square root symbol. Therefore, the square root of 48 simplifies to 4âˆš3.

Let's try another example. Simplify the square root of 450. By writing out a factor tree, you can see that 450 can be factored to (3 * 3) * (5 * 5) * (2). As you can see below, there is one pair of 3s and one pair of 5s. The pair of 3s and pair of 5s, when multiplied together, equal 225, which has a perfect square root of 15. This means that a 15 will be removed from the square root symbol. Because the 2 does not have a partner, it will remain under the square root symbol. Therefore, the square root of 450 simplifies to 15âˆš2.

The proper mathematical way to write imperfect square roots is to simplify them as much as possible without using a fraction or decimal. This is done by factoring the number as much as possible and then moving those numbers with a partner. This can be done because the square root of a number is defined as a number that, when multiplied by itself, gives you the original number. Any number without a partner cannot be simplified and will remain under the square root symbol.

After you've completed this lesson, you'll have the ability to:

- Differentiate between perfect squares and imperfect squares
- List the steps to evaluate and simplify an imperfect square

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ELM: CSU Math Study Guide17 chapters | 147 lessons | 7 flashcard sets

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