If a** **is a real number and

*n*is a natural number, then every solution of the equation

**a af**

*xⁿ =***r x in the set R**

*te***if any, is called th**

*,***nth root of**

*e***.**

*a*The root is denoted (read:* *nth root of* a*). The numbe** r** n is calle

**d the root pointer, the**number a is

*t*

**he sub-root expression,**and the symbol √ is t

**he root characte**r.

When ** n** is an odd number, it makes sense for any a

**and when**

*∈ R,***is an even number, then the symbol only makes sense i**

*n**f a.*0.

The procedure by which it is determined is ca**lled rootin**g.

The following is true of the root:

when *n* is an odd number;

when n* *is an even number.

## 1. EXPANDING AND SHORTENING THE ROOTS

1) If a* *is a positive real number and* m, *n an*d* p are natural numbers, then

This property is called root extension.

2) If the roots are given and there is a com*mo*n d*i*visor k for the numbers m a*nd* n, then:

This property is cal**led root shortening.**

## 2. RADICATION OF PRODUCT AND QUANTITY

1) If a** **and

**b are positive real numbers, then for every natural number**

**the equation holds:**

*n***Note**. *The rule for multiplication of roots by equal root indicators is given by the law.*

If two or more multiplicative roots have different root indices, they should first be reduced to roots with equal indices.

2) If a** a**n

**b are positive real numbers, then for every natural number**

*d***, the equation is true:**

*n*Note. Pawenity

gives the rule for dividing the root by equal root pointers.

## 3. HOPMAL TYPE OF ROOT

### 3.1. Extract the multiplier before the root sign

If the root is given, where m** > n a**n

*0, then*

**d m = np + q**, and>In that case, ap is extracted before the sign of the r*oo*t, and aq thnnuvn in the *roo*t, ie.

### 3.2. Enter a multiplier before the root sign

The procedure for entering a multiplier under the sign of the root is the opposite of that for extracting a multiplier before the sign of the root, ie. if there is a multiplier in front of the root to enter under the root sign, it should be scaled with the root pointer first.

### 3.3. Normal root type

A root is said to be in the normal form (simplified form) when the rooted expression:

– does not contain a denominator;

– does not contain multipliers that can be extracted before the root sign;

– when the root indicator and the sub-root expression indicator do not have a common divisor.

**Example.**

## 4. ROOT GRADUATION

If a* ∈ R +* an*d n, p ∈ *N, then the equation is true:

## 5. ROOT ROOTING

If *a ∈ R +* an*d n, p ∈ *N, then the equation is true:

## 6. DEGREE WITH INDICATOR RATIONAL NUMBER

If m and n are natural numbers and if a is a positive number, then:

M / n *is c*alled a d**egree with a rational indicator.**

## 7. IRRATIONAL EXPRESSIONS

The expressions in which it is represented and the rooting are calle**d irrational expression**s.

Irrational expressions of the ty*pe *AJV, where A* a*nd *B* are rational expressions, are calle**d root**s. The expression A is call**ed the coefficient of the roo**t.

Two or more roots that are brought to normal form are called **similar root**s if they have equal sub-root expressions and equal root pointers.

### 7.1. Collection and removal of roots

These operations are performed with the help of transformation reduction to similar roots.

Example:

**7.2. Multiplication of irrational expressions**

The multiplication of irrational expressions is done in the same way as the multiplication of rational expressions, using the rule for multiplication of roots.

Example:

**7.3. Sharing irrational expressions**

The division of irrational expressions is done as well as the division of rational expressions, using the rule of division of roots.

**Example.**

**7.4. Rationalize the fraction denominator**

The transformation by which the denominator of a fraction of an irrational expression is converted into a rational expression is cal**led rationalizing the denominator of the fraction.**

1) Rationalize the denominator of the fraction of the type

2) Rationalize the denominator of the fraction of the type

3) Rationalize the denominator of the fraction of the type