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- Square numbers

- About
# Square/cube root calculator

In the exact sciences, the operation of extracting the square root is widely used. It is denoted by the sign "√" and is calculated as squaring "vice versa". That is, x = √x². In this case, x must be a real number, or complex - in the case of a mathematical generalization. Since x and −x squared give the same positive value, there is the concept of an arithmetic square root, whose value is always ⩾ 0.

## History of the square root

Archaeological studies indicate that the square root was used as early as the 18th-16th centuries BC - in the Babylonian kingdom. Direct proof of this is the YBC 7289 tablet found in Babylon, dated 1800-1600 BC. It shows the results of extracting the square roots of 2 and 2/2, to five decimal places.

In ancient Greece, around the 4th century BC, Theaetetus of Athens discovered an important regularity of the square root. If it is impossible to extract an integer (natural) root from a number, the result will always be irrational, that is, it cannot be represented as an ordinary fraction m / n, where m and n are integers. So, if √4 = 2, then √3 = 1.7320508075688772935... A vivid example of such irrational numbers is the constant π with an infinite number of non-repeating digital combinations after the decimal point.

Then, in antiquity, the Latin name for the square root appeared - "radical", which translates as "root" (radix). In the Middle Ages, this was the reason for the emergence of a new designation / abbreviation for square roots - Rx. It, among other things, was used in his writings by the Italian mathematician Gerolamo Cardano. And the modern designation of the square root - "√" was first introduced by the German scientist Christoph Rudolph (Christoph Rudolff) in 1525. Initially, there was no dash above the radical expression, and it began to be used everywhere only in 1637 - after the publication of René Descartes' Geometry.

From the 16th to the 17th century, research on the radical in Europe showed that it could be used in the mathematics of imaginary numbers. Rafael Bombelli developed a way to calculate them using continued fractions, and Abraham de Moivre proved that taking the square root of any complex number does not lead to new types of numbers. And in the 19th century, Carl Friedrich Gauss conducted a deep study of complex roots of an arbitrary degree, opening up new horizons for scientists for further scientific research.

Today, the square root is widely used in geometry, physics, numerical methods, celestial mechanics, probability theory, statistics and computer science. For example, in many functional-level programming languages, "√" is denoted as sqrt - from the English square root.

## Cube root

In mathematics, among all the existing degrees, the most important role is played by the "square" and "cube", found in most formulas. Accordingly, the most necessary roots are square and cubic. If the first can be represented as x = √x², then the second is calculated as x = ³√x³.

A minus times a minus gives a plus, which means that the result of taking the square root will always be positive. But this rule does not work in the case of the cube root, which is an odd function. Therefore, for its calculations, a "real" (modular) result is provided:

³√x = ³√−x.

There is another form of expressing roots: through fractional degrees. According to her, the square root of the x can be written as x¹/², and the cubic root - x¹/³. In the case of complex numbers, the following equation will be true:

x¹/³ = exp ((1/3)ln x),

where ln is the principal value of the natural logarithm.

Without delving into higher mathematics, it is worth noting that the cube root is indispensable in calculations related to the volumes of bodies. Including - completely or partially submerged in water: ships, ships, submarines, bathyscaphes and so on. Many formulas describing the operation of water / underwater transport use the third power (x³) and the cube root (³√x). The same applies to air transport and spacecraft that go beyond the earth's atmosphere.

The interesting facts also include the following: unlike the square root, the cube root (in geometry) cannot be extracted with the help of "improvised" measuring devices: rulers and compasses. Therefore, such geometric actions as doubling a cube, building a regular heptagon, trisection of an angle, and so on are incompatible with it.

At the same time, the dimensions of all bodies with the same density are related to each other as the cube roots of their masses. So, if one round body weighs 2 times more than the other, the difference in their diameter will be only 26%. This often causes a deception of perception, for example, when comparing celestial bodies. So, visually (in diameter) the Earth is only 3.7 times larger than the Moon, but at the same time it is 84 times heavier.

- Help
## How to find square/cube root

Raising numbers to powers and extracting roots from them are algebraic operations, without which it is difficult to imagine modern exact sciences, including mechanics, optics, astronomy, computer science. They are most in demand in geometry: when calculating flat and three-dimensional figures, their interaction with each other and being in space.

### Square Root Properties

The most common arithmetic root is the square root, or √a = x, which can also be expressed as a quadratic equation: x² = a. It has two important restrictions: x and a must not be less than zero (x ⩾ 0, a ⩾ 0). Before calculating equations with square roots, they are simplified as much as possible using special formulas/properties:

√a⋅b = √a ⋅ √b.

In text form, this property looks like this: "the root of the product is equal to the product of the roots." In this case, a and b must not be less than zero. For example, substituting the number 4 for a, and the number 9 for b, we get:

√4⋅9 = √4 ⋅ √9 = 2 ⋅ 3 = 6.

In this case, the root is extracted even if a and b are first multiplied together: √4⋅9 = √36 = 6. But this is a special case, taken as an example, and in real calculations such “convenient” and "correct" numbers. For example, if you substitute 23.6 and 655.9 instead of a and b, respectively, the task becomes feasible only using a calculator.

The second important property is the extraction of the root of a fraction, which is always equal to dividing the root of the numerator by the root of the denominator:

√(a/b) = √a / √b.

The restriction here is not the same as in the previous rule, namely: a ⩾ 0, b > 0. Substituting the number 16 for a and 9 for b, we get:

√(16/9) = √16 / √9 = 4 / 3, or ~1.33.

The third rule is about degrees. So, to raise the root to a power, you need to raise the root expression to the power:

(√a)ᵇ = √aᵇ.

In this case, a must be greater than or equal to zero. Substituting the number 4 for a and the number 2 for b, we get:

(√4)² = √4² = √16 = 4.

All three rules are also relevant for the cube root, that is:

³√a⋅b = ³√a ⋅ ³√b, ³√(a/b) = ³√a / ³√b, (³√a)ᵇ = ³√aᵇ.

If there are simple natural numbers instead of unknowns, it is highly likely that you can extract the root yourself: without a calculator. To do this, just use the methods presented below.

### Extraction of roots

The main principle in solving any equation is simplification, or decomposition of one complex problem into several smaller ones. In the case of square roots, simplification can be done in several ways:

#### Decomposition of the root expression into factors

A complex number can often be expressed as a product, such as 400 as 4 ⋅ 100, 2 ⋅ 200, 8 ⋅ 50, and so on. In many cases, this helps solve the root extraction problem. By placing the number 400 under the sign "√", we represent it in the form:

√400 = √25⋅16 = √25 ⋅ √16 = 5 ⋅ 4 = 20.

The ultimate goal of the expansion of the radical expression is to find possible square numbers, that is, numbers from which the square root can be taken without a trace. For example, the solution described above would not work if 400 were represented as 2 ⋅ 200 or 8 ⋅ 50, but it would work if using the form 4 ⋅ 100, since 4 and 100 are square numbers.

#### Removing one square number as the root

In most cases, the radical number cannot be decomposed into several square numbers, but it remains possible to extract at least one such number. For example:

√147 = √49⋅3 = √49 ⋅ √3 = 7√3.

This is one of the basic principles of simplifying equations, which can help you find the approximate value of the root without a calculator. So, √147 is impossible to calculate mentally, but 7√3 is quite realistic, if you remember that √3 is approximately equal to 1.73.

#### Comparing the value of the root with "adjacent" extractable roots

This is the easiest and most visual way to extract square roots in your mind, and with a fairly high accuracy: up to decimal fractions. For example, if you need to calculate √35, take the nearest "extractable" square numbers: 25 and 36.

√25 = 5, √36 = 6.

So the desired root of 35 is between these two values. Since 35 is much closer to 36 than to 25, we find an approximate answer: √35 = 5.9. An accurate calculation on the calculator will confirm the result: 5.92.

For small and simple root expressions, these tips can really help. But when it comes to large numbers, the task becomes not only difficult, but impossible. Then it doesn't make sense to rack your brains and it's easier to use either an engineering calculator or a special online application that will give you the result in a split second!

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