Rationalization of Denominators
The rationalization of denominators is a mathematical concept that is related to the simplification of fractions. It consists of a technique for finding an equivalent expression of a fraction, either by eliminating roots or by factoring at the bottom.
How is the rationalization of denominators carried out?
- 1. Solving square roots:
- For example, if you have a fraction of the form: $$frac{a}{sqrt{b}}$$ This can be simplified by using the square of the denominator: $$frac{a}{sqrt{b}} cdot frac {sqrt{b}}{sqrt{b}} = frac{a cdot sqrt{b}}{b}$$
- 2. Development of the factor on the denominator:
- For this, it is important to know some properties of prime numbers:
- if $$a divides b divides c implies a divides c$$
- if $$a divides b times c implies a divides b text{ or } a divides c$$
- if $$a divides b times c implies a^2 divides b^2 times c$$
- For this, it is important to know some properties of prime numbers:
It is also important to know the common factor theorem, which tells us that, if there is a common factor in two or more terms, it can be taken outside the parentheses, that is, it was taken next to the numerator.
Practical examples
- Simplify the following fraction: $$frac{2}{sqrt{8}-sqrt{15}-sqrt{12}}$$
- First, we solve for square roots in the denominators:
- $$frac{2}{sqrt{8}-sqrt{15}-sqrt{12}} cdot frac{sqrt{8}+sqrt{15}+sqrt{12}}{sqrt{8}+sqrt{15}+sqrt{12}} =frac{2(sqrt{8}+sqrt{15}+sqrt{12})}{8-15-12}$$
- Next, the denominator factors are developed:
- $$frac{2(sqrt{8}+sqrt{15}+sqrt{12})}{8-15-12}=frac{2(sqrt{8}-sqrt{3}+3sqrt{2})}{-21} $$
- Finally, the factors are taken to the numerator:
- $$frac{2(sqrt{8}-sqrt{3}+3sqrt{2})}{-21}= frac{-2(sqrt{24}-3sqrt{3})}{-21}=frac{sqrt{24}-3sqrt{3}}{21} $$
As can be verified, using the denominator rationalization technique it was possible to simplify the given fraction, by remaining with $$frac{sqrt{24}-3sqrt{3}}{21}$$
Rationalization of Denominators
What does it mean to rationalize the denominators?
Rationalizing the denominators means removing radicals from the denominators. It is presented as a way to simplify operations and equations. By removing radicals and obtaining simplified fractions, simpler operations are achieved.
When to use denominator rationalization
We can use the rationalization of denominators when we work with fractions whose operations have radicals in their denominator. By removing radicals in the denominator it will allow us to operate with simple fractions. This leads to less error in the operation process.
How is the denominator rationalized?
To rationalize the denominators, some steps must be followed:
- Remove factors from the denominator: First we must remove the common factors of the denominator to simplify the expression.
- Multiply both numerator and denominator: Next we multiply both numerator and denominator with the content of the radical of said denominator.
- Simplify the resulting expression: Finally, we have a new rationalized fraction where radicals have been removed from the denominator.
Examples
- Rationalize the denominator of 2/√(5)
- Solución: $$frac{2*sqrt{5}*2}{sqrt{5}*2*sqrt{5}} = frac{4*2}{2*5} = frac{8}{10}$$
- Rationalize the denominator of 4/√(3)·√(2)
- Solución: $$frac{4*sqrt{6}}{sqrt{3}*sqrt{2}*sqrt{6}} = frac{4*sqrt{6}}{2*3*sqrt{6}} = frac{4*sqrt{2}}{6}$$
Conclusion
Rationalizing denominators simplifies fractions and equations by removing radicals from the denominator. With the purpose of reducing errors in operations. To achieve this, it is necessary to follow certain steps where you can remove the factors from the denominator along with multiplying both the numerator and denominator with the content of the radical thereof, and then simplify the fraction.