For all real numbers a and b ,. Since a negative number times a negative number is always a positive number, you need to remember when taking a square root that the answer will be both a positive and a negative number or expression. The idea here is to find a perfect square factor of the radicand, write the radicand as a product, and then use the product property to simplify.
If the number under the radical has no perfect square factors, then it cannot be simplified further. For instance the number 17 cannot be simplified further because the only factors of 17 or 17 and 1.
So, there are no perfect square factors other than 1. An expression is considered simplified only if there is no radical sign in the denominator. Simplifying the square roots of powers. Fractional radicand. A radical is also in simplest form when the radicand is not a fraction. Example 1. Therefore, is in its simplest form. Example 2. Extracting the square root. Therefore, is not in its simplest form. We have,. The radicand no longer has any square factors. The square root of a product is equal to the product of the square roots of each factor.
We will prove that when we come to rational exponents , Lesson Here is a simple illustration:. All of these radical terms involve multiplication, not division. Therefore we should try to simplify each expression using the product property of roots.
A radical term can be simplified using this property if its radicand has a factor that is also a perfect square. Therefore this radical term can be factored.
We can prove this by factoring it:. You can find the side length of the square using the area of the square, and the fact that the area is equal to the square of the side length.
The small number is called the index. It indicates which root is being taken. It is equal to two in the case of a square root. The different expressions sometimes share common factors. It may help to start by simplifying each radical expression without looking at the answers.
Each expression can be simplified using the product property of roots. Let's simplify each radical expression. We are using cookies and various third-party services to optimize our platform for you, to continually improve our content and offerings for you and to measure and manage our advertising. You can find detailed information in our privacy statement.
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JavaScript is deactivated for your browser. Fill gaps in your understanding — without pressure. Already have an account? Email address. I forgot my password. Simplifying Radical Expressions. Sign up quickly and start your FREE trial. Rate this video. To Google. The author. Susan Sayfan. Description Simplifying Radical Expressions For a radical expression to be in the simplest form, three conditions must be met: 1. Radicals , like fractions, can be simplified until you have the smallest possible parts.
For starters, lets look at a simplified perfect square. This can also be done with numbers that aren't perfect squares. Let's look at a square root that will have some steps to simplifying.
We can use the multiplication and division properties of radicals to do that! Be mindful of what numbers are inside of the radical sign and which are not! Unless you can take the square root of the individual prime numbers, they must remain under the radical. You can multiply different variables under the radical to simplify.
First we're gonna separate the radical into parts to make it easier to work with, then we can simplify the parts. Now what if the numbers under the radical aren't perfect squares?
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