## Radicals theory

# Radicals

Radicals are expressions with the form:

Where n Є N and a Є R; provided that when a is negative, n must be odd.

n=index; a=radicand; m=exponent.

A radical can be expressed in the form of power, and therefore power radical as follows:

Using the notation of fractional exponent and property of the fractions that says if we multiply numerator and denominator by the same number, the fraction is equivalent, we obtain an equivalent radical to the original one.

In other words, if you multiply or divide the index and the exponent of a radical by the same natural number, an equivalent radical is obtained.

If there is a natural number that divides the index and the exponent of the radicand, a simplified radical is obtained.

How to reduce into a common index?

First we find the least common multiple of the indices, which will be the common index. Then we divide the common index for each of the indexes and each result is multiplied by their corresponding exponents.

Example:

To remove factors of a radical, the radicand is decomposed in factors. If:

- An exponent is less than the index, the corresponding factor is left in the radicand.

- An exponent is equal to the index, the corresponding factor goes out of radicand.

- An exponent is greater than the index, this exponent is divided by the index. The coefficient obtained is the exponent of the factor outside the radicand and the rest is the exponent of the factor in the radicand.

To introduce factors in a radical, we elevate it to the correspondent index.

Regarding the addition or subtraction of radicals, they can only be added (or subtracted) when two radicals like similar radicals, that is, if they are radicals with the same index and the same radicand.

To multiply or divide radicals, it must be taken into account that if they are the same or different index. In the case of multiplications with the same index, radicands are multiplied and the same index is allowed.

In the case of division, the radicands are divided and it is left the same index.

If, however, the roots have different index, they should be reduced to common index and then multiply or divide them as the operation to perform.

When we have a radical raised to a power, we must raise the filing, leaving the same index root.

If, however, the operation that we perform is a root of a radical, the result is another equally radical radicand and whose index is the product of the two indices.

What is streamlining?

Radical streamlining involves removing radicals denominator, allowing easy calculation operations such as addition (or subtraction) of fractions.

We can distinguish three cases:

1.- Rationalization type:

The numerator and denominator are multiplied by

2.- Rationalization type:

The numerator and denominator are multiplied by

3.- Rationalization type:

And in general when the denominator is a binomial with at least one radical. The numerator and denominator are multiplied by the conjugate of the denominator. The conjugate of a binomial is equal to the binomial with the central sign changed:

We also need to bear in mind that: sum by difference is equal to difference of squares.

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