How to Simplify Non Perfect Roots
Non perfect roots, also known as irrational roots, can be quite challenging to simplify. However, with the right approach and understanding of the properties of square roots, you can simplify these roots with ease. In this article, we will discuss the steps and techniques required to simplify non perfect roots.
Understanding the Concept
Before diving into the simplification process, it is essential to understand the concept of non perfect roots. Non perfect roots are those that cannot be expressed as a fraction, and their decimal representation is non-terminating and non-repeating. An example of a non perfect root is the square root of 2 (√2).
Prime Factorization
The first step in simplifying a non perfect root is to express the number under the square root as a product of its prime factors. Prime factorization involves breaking down the number into its prime components. For instance, to simplify √54, you would first factorize 54 into its prime factors: 54 = 2 × 3 × 3 × 3.
Grouping Prime Factors
Once you have the prime factors, the next step is to group them in pairs. If a prime factor appears an odd number of times, leave it as it is. For example, in the prime factorization of 54, we have three 3s, so we group them as (3 × 3) × 3.
Extracting Square Roots
After grouping the prime factors, you can now extract the square roots. For each pair of prime factors, take one factor from each pair and place it outside the square root. For our example, √54 = √(2 × 3 × 3 × 3) = √(2 × (3 × 3) × 3) = √(2 × 9 × 3) = √(18).
Further Simplification
If the number under the square root is still not simplified, you can continue the process by factoring it further. In our example, √18 can be simplified further by factoring out the perfect square 9: √18 = √(9 × 2) = √9 × √2 = 3√2.
Conclusion
Simplifying non perfect roots may seem daunting at first, but with a clear understanding of prime factorization and the properties of square roots, you can simplify these roots with ease. By following the steps outlined in this article, you will be able to simplify non perfect roots and gain a deeper understanding of the properties of square roots.
