Arithmetic roots of natural degree

The root of the second degree (square root) of a is the number that becomes equal to a if it is raised to the second degree (squared).

Example 1

 – number 8 is the second root of 64

 – the number 0.6 is the root of the second degree of 0.36

 – number 1 is the root of the second degree from number 1

Do not forget to mention that there are numbers for which it is impossible to find a square equal to this number, which would be a real number. Simply put, not for all numbers you can find a real number whose square would be equal to a given number.

Remark 1

For any number  at a negative rate  not true because  cannot have a negative value for any indicator .
The conclusion follows: for real numbers, there is no square root of a negative number.

Because the , then zero is the square root of the number “zero”.

Definition 2

The arithmetic root of the second power of the number  Is a non-negative number that becomes equal if you square it.

Arithmetic root of the second degree from among  has the following designation: . However, there is such a designation:, but the deuce (root index) does not need to be prescribed.

Sign of arithmetic root ““Also has the name” radical “. It should be remembered that the “root” and “radical” are complete synonyms (they have exactly the same meaning and are used both in that and in that version).

The number under the root sign is the root number . If the whole expression is under the sign of the root, then it is customary to call it the root expression , respectively.

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Definition 3

Looking at the definition of the concept of ” arithmetic root “, we can derive the following formula:

For anyone 

The word “arithmetic” when reading a record  can be omitted.

Further we will consider exclusively arithmetic roots from non-negative numbers and expressions.

Cubic root

Definition 4

The arithmetic root of the third degree (cubic root) is a non-negative number, which, if it is cubed, will become equal. Designated as.

Number  in this entry, the root metric . The number or expression under the root sign is the root .

Again, the word “arithmetic” is most often not used, but simply said: “the third root of the number ».

Example 2

– arithmetic root of the 3rd degree from  or cubic root of ;

– arithmetic root of the 3rd degree from  or cubic root of .

Arithmetic root of the nth degree

Definition 5

The arithmetic root of the nth degree from a≥0a≥0 – non-negative number, which, subject to exponentiation, becomes equal to the number  and is indicated by: where  – the radical number or expression, and  – an indicator of a root.

The arithmetic root can be written using the following characters:

.

Example 3

 – arithmetic root of the seventh degree from among where  Is the root number, and  – an indicator of a root.

 – arithmetic root of  Where  Is the root expression, and  – an indicator of a root.

Based on the definition of the arithmetic root -th degree, the radical expression must be a non-negative number or expression. If in equality both parts multiplied by , then we get two equivalent parts of the equality: 

From this it follows that for odd exponents of the arithmetic root, the following equality is written:

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