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Using Square Root Symbols to Show Solutions to Rational Equations

Instructor: Maria Blojay

Maria has taught College Algebra and has a master's degree in Education Administration.

This lesson is about using square root symbols to show solutions to rational equations. We will learn about the difference between positive and negative square roots, radical expressions, and how to evaluate them.

What is a Square Root?

When a number is multiplied by itself, the product is equal to the square of the number. Another way of saying this is, any number that is the product of repeated whole-number factors is a 'perfect power.'

For example:

Eqn1

Suppose we are standing inside a square room with each side measuring 10 feet. The room has an area of 100 square feet.

Since the length and width of the square room have the same measurement, 10 feet, then:

Area = length times width

Eqn2

When you look for a number that you can multiply by itself to get a product, you are looking for the 'square root' of the product.

Simply, the square root symbol is called a 'radical':

Eqn3

The radicand can be a variable or expression.

The square root of one hundred forty-four equals twelve. This is written in math as:

Eqn4

or we could ask what number can you multiply by itself two times to get 144? The number is 12.

So, let's take a look at this problem:

Eqn5

We ask ourselves, what number multiplied by itself will give us 121? The answer would be 11 because 11 times 11 = 121.

Positive and Negative Square Roots

Note, that square root means the 'principal' square root of a number, which is the positive root.

However, each positive number also has a 'negative' square root.

If you multiply two negative numbers, you also get a positive product.

(-7)(-7) = 49

This means that -7 is also a square root of 49. However, square root means the principal square root, which is the positive root.

If you're asked to find x when

Eqn6

See how the + sign is stacked on top of the - sign, to indicate both a positive and negative square root.

Let's explore this problem to find:

Eqn7

Remember to think what number can we multiply by itself two times will give us 1600?

So, 40 times 40 = 1600.

Our final answer is:

Eqn8

Let's look at this one:

Notice here that after taking the square root, the original problem wants the positive and negative values by using the symbols:

PM

Eqn10

Relationship Between Squares and Square Roots

There is a square root property:

Eqn11

which says that the square root of a number b 'squared' or raised to the 2nd power will give you that number b back.

For example:

Eqn12

Think of it this way.

Eqn13

We got the same number back based on the square root property.

Eqn14

So, we could choose any number, let's say 45:

Eqn15

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