4.3 Evaluating and simplifying radicals

It was said that he radical sign was first used by Rene Descartes, but long before he used this, it has been developed in Germany in the 1500s. The radical ( sign was invented to find the square root or the cube root, 4th root, 5th root etc., depending on the index of the radical sign.

Now if you remember, in previous chapter, we learned all about the different number systems and we have mentioned about the rational and irrational numbers. Radicals can belong to either of which depending of course if the radicals would be able to fit into the definition of whichever group of number. In this lesson we will get to be able to identify whether or not a certain radical is rational. We will be specifically going to study about a square root or a cube root of a particular number.

We will first look into what a square and a cube are since these would be the two fundamental concepts. In order to solve a square root, or a cube root, one must know how to square or cube a number. These will be thoroughly discussed in 4.1 and 4.2.

For 4.3 to 4.8 we will be looking closer into how to simplify and evaluate these radicals through applying what we have learned from chapters on square and square roots, estimating square roots, and rational numbers from earlier grades. We would also be learning how to convert them into either entire radicals to mixed radicals or vice versa, by applying our knowledge of factoring out the perfect squares or the perfect cubes in the equation. Apart from converting the radicals from one form to another we are also going to learn more about how to combine different radicals and simplify them depending on the operation used and learn to rationalize a certain radical equation.

Evaluating and simplifying radicals

To simplify radicals, we will need to find the prime factorization of the number inside the radical sign first. Then, move each group of prime factors outside the radical according to the index. The last step is to simplify the expression by multiplying the numbers both inside and outside the radical sign.


  • 1.
    Evaluate the radicals
  • 2.
    Using calculator to evaluate
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Evaluating and simplifying radicals

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