# A negative number is a perfect square

## Calculate with square and cube roots

### Video: Calculating with square and cube roots

In this tutorial we deal with the question of what actually a **Square-** or **Cube root** is.

### The pulling of the roots - repetition

The extraction of roots is the opposite of exponentiation. Instead of pulling roots, one also says rooting. The mathematical symbol for extracting the root is the root sign: $ \ sqrt {\ textcolor {white} {...}} $

$ Exponentiate ~~~~~~~~~~~~~~~~~~~~~ Square root $

$ \ textcolor {red} {8} ^ \ textcolor {blue} {2} = \ textcolor {green} {64} ~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~ \ sqrt [\ textcolor {blue} {2}] {\ textcolor {green} {64}} = \ textcolor {red} {8} $

*spoken: The second root of sixty-four is eight*.

A **root** is given by:

$ \ textcolor {red} {a} ^ \ textcolor {blue} {n} = \ textcolor {green} {x} \ Leftrightarrow \ sqrt [\ textcolor {blue} {n}] {\ textcolor {green} {x} } = \ textcolor {red} {a} $

Here $ \ textcolor {green} {x} $ is called the radicand and $ \ textcolor {blue} {n} $ is the root exponent.

### What is a square root?

**Roots whose root exponent is $ 2 $ are also known as square roots.** As a rule, however, the root exponent $ 2 $ is omitted because the square root is the most common root. All root exponents that are greater than $ 2 $ must always be added.

$ \ sqrt [2] {49} = 7 \ Leftrightarrow \ sqrt {49} = 7 $

$ \ sqrt [2] {9} = 3 \ Leftrightarrow \ sqrt {9} = 3 $

$ \ sqrt [2] {16} = 4 \ Leftrightarrow \ sqrt {16} = 4 $

The square root of a number is the number that squares the number below the root. So a general form of the square root would look like this:

If $ x \ cdot x = y $, then: $ \ sqrt {y} = x $

Since taking the square root results in crooked numbers in many cases, you will mostly calculate them with the help of a pocket calculator. In these cases you need to round the numbers:

- $ \ sqrt {2} \ approx 1.4142 $

- $ \ sqrt {10} \ approx 3.1623 $

The **square root** is described by:

$ \ textcolor {red} {x} ^ \ textcolor {blue} {2} = \ textcolor {green} {y} \ Leftrightarrow \ sqrt {\ textcolor {green} {y}} = \ textcolor {red} {x} $

The square root can also be extracted from fractions by taking the root individually from the numerator and denominator.

- $ \ sqrt {\ frac {4} {9}} = \ frac {\ sqrt {4}} {\ sqrt {9}} = \ frac {2} {3} $

- $ \ sqrt {\ frac {16} {4}} = \ frac {\ sqrt {16}} {\ sqrt {4}} = \ frac {4} {2} = 2 $

Due to the relationship between square root and squaring (take it up with two), you cannot take a square root from a negative number.

$ x ^ 2 = y \ sqrt {y} = x $

$ x ^ 2 $ can never result in a negative number:

$(-2)^2 = 4$

$2^2 = 4$

It's math **Not** possible to extract square roots from negative numbers!

### What is a cube root?

The cube root differs from the square root in that it has the root exponent.

$ Exponentiate ~~~~~~~~~~~~~~~~~~~~~ Square root $

$ \ textcolor {red} {3} ^ \ textcolor {blue} {3} = \ textcolor {green} {27} ~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~ \ sqrt [\ textcolor {blue} {3}] {\ textcolor {green} {27}} = \ textcolor {red} {3} $

Instead of taking the number to the power of two, the cube root asks for numbers that were taken to the power of three. Another difference to the square root is that you always have to write the $ \ textcolor {blue} {3} $ at the root and not simply leave it out.

$ 4 ^ 3 = 64 \ rightarrow \ sqrt [3] {64} = 4 $

$ 9 ^ 3 = 729 \ rightarrow \ sqrt [3] {729} = 9 $

$ 1 ^ 3 = 1 \ rightarrow \ sqrt [3] {1} = 1 $

The **Cube root** is described by:

$ \ textcolor {red} {a} ^ \ textcolor {blue} {3} = \ textcolor {green} {x} \ Leftrightarrow \ sqrt [\ textcolor {blue} {3}] {\ textcolor {green} {x} } = \ textcolor {red} {a} $

In contrast to the square root, the cube root can also be obtained from negative numbers:

$(-3)^3 = - 27$

$ \ sqrt [3] {- 27} = -3 $

### How do I write a root as a power? - Method

A *root* presses a **reverse potency** out.

Roots and powers are very closely related. For one thing, a root practically expresses an inverse power. On the other hand, roots can also be written as powers:

$ \ sqrt [3] {64} = 64 ^ {\ frac {1} {3}} = 4 $

$ \ sqrt [2] {81} = 81 ^ {\ frac {1} {2}} = 9 $

Roots can be written as powers by taking the radicand as the base and the reciprocal of the root exponent as the exponent:

$ \ sqrt [\ textcolor {blue} {n}] {\ textcolor {green} {x}} = \ textcolor {green} {x} ^ {\ frac {1} {\ textcolor {blue} {n}}} $

Your newly learned **Knowledge** to square roots and cube roots you can now with our **Exercises** testing! We wish you a lot of success and fun!

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