How do radicals

In each case, we check our answer by substituting it in the original equation. For example, in the last equation we obtain:. Later in the course we will consider more complicated cases of radical equations.

Numerical Values The radicals in the above examples were all natural numbers. This is due only to a judicious choice of examples. Frequently the roots occurring in applications are irrational numbers with decimal expansions that never repeat or terminate. We will need to do a little more work before we can deal with the last two. In this case the exponent 7 is larger than the index 2 and so the first rule for simplification is violated.

To fix this we will use the first and second properties of radicals above. The radical then becomes,. Now use the second property of radicals to break up the radical and then use the first property of radicals on the first term.

To do this we noted that the index was 2. We then determined the largest multiple of 2 that is less than 7, the exponent on the radicand.

This is 6. In the remaining examples we will typically jump straight to the final form of this and leave the details to you to check. This radical violates the second simplification rule since both the index and the exponent have a common factor of 3. To fix this all we need to do is convert the radical to exponent form do some simplification and then convert back to radical form. Although, with that said, this one is really nothing more than an extension of the first example.

There is more than one term here but everything works in exactly the same fashion. We will break the radicand up into perfect squares times terms whose exponents are less than 2 i. Now, go back to the radical and then use the second and first property of radicals as we did in the first example.

Note that we used the fact that the second property can be expanded out to as many terms as we have in the product under the radical. This will happen on occasion. This one is similar to the previous part except the index is now a 4. So, instead of get perfect squares we want powers of 4. This time we will combine the work in the previous part into one step.

That will happen on occasion. This last part seems a little tricky. Individually both of the radicals are in simplified form. Radicals, also called roots, are the opposite of exponents.

Simplifying radical expressions uses many of the same tricks you've learned in earlier math lessons for simplifying fractions or exponents. If you're new to the topic, start by learning how to simplify the square root of an integer. To simplify a radical expression, simplify any perfect squares or cubes, fractional exponents, or negative exponents, and combine any like terms that result. If there are fractions in the expression, split them into the square root of the numerator and square root of the denominator.

If you need to extract square factors, factorize the imperfect radical expression into its prime factors and remove any multiples that are a perfect square out of the radical sign. For tips on rationalizing denominators, read on! Did this summary help you? Yes No. Log in Social login does not work in incognito and private browsers. Please log in with your username or email to continue.

No account yet? Create an account. Edit this Article. We use cookies to make wikiHow great. By using our site, you agree to our cookie policy. Cookie Settings. Learn why people trust wikiHow. Download Article Explore this Article methods. Tips and Warnings. Related Articles. Article Summary. Method 1. All rights reserved. This image may not be used by other entities without the express written consent of wikiHow, Inc. Factor the number under the square root.

Ignore the square root for now and just look at the number underneath it. Factor that number by writing it as the product of two smaller numbers. If the factors aren't obvious, just see if it divides evenly by 2. If not, try again with 3, then 4, and so on, until you find a factor that works.

The first step is finding some factors of Keep going until the number is factored completely. Remember, any number can be factored down into prime numbers like 2, 3, 5, and 7. Keep breaking down the factors until there are no more factors to find.

Rewrite pairs of the same number as powers of 2. Varsity Tutors connects learners with experts. Instructors are independent contractors who tailor their services to each client, using their own style, methods and materials. Simplifying Radical Expressions Before you can simplify a radical expression, you have to know the important properties of radicals.

Example 1: Simplify. Example 2: Simplify. Example 3: Simplify. Example 4: Simplify.