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Square Root

Square Root

The square root is an operation to determine the two same factors of a number. You root a number by evaluating the same factors.

  Example

The square root of 9 is 3, because the product of 3 with 3 results 9.

The notation of a square root is a stretched out lower r

  Example

But:

, for  or

The square root of a given number is always positiv. In opposition to this fact the square root of an unknown number x² could be positiv or negativ.

Attention!

The square root of a negative number is not a real number, but an imaginary number. This number has the notation i.

Another writing of square roots is a broken number, a fraction, as exponent.

The denominator 2 indicates the square root. The numerator is the exponent of the number under the root sign. In this example it is 1.

If you multiply  by itself you get x, for .

To find out the square root of a number, you can factor the number. The even powers are important. They can be square rooted. If there is an odd exponent, the even exponent of it can be square rooted, the rest remain behind the root sign.

  Examples

Only even exponents

Even and odd exponents

The number 2 consists of 3 factors. Two factors can be square rooted. The third factor must remain behind the square sign. The exponent of the number 3 is even, so it can be square rooted.

Multiplication rule

Several multiplicated square roots can be put together to one root. Vice versa a root can cleverly be splitted to extract a factor with an even exponent.

  Examples

Put together to one square root:

Extract the apted square root:

Division rule

You can write a root of an entire fraction  or you can separate the square root into the root of the numerator and into the root of the denominator .

  Examples

Separate the entire root to single roots and factor then:

Remark: Usually a mathematician put a root into the numerator. A root shall not remain in the denominator. So the equation would be solved, if you multiply the root in the denominator with the numerator and with itself.

Now it is complete!