The concept of degree in mathematics is introduced in the 7th grade in an algebra lesson. And later on, throughout the entire course of studying mathematics, this concept is actively used in its various forms. Degrees are a rather difficult topic that requires memorizing values and the ability to correctly and quickly count. For faster and better work with degrees, mathematicians came up with the properties of the degree. They help reduce large computations, transform a huge example into a single number to some extent. There are not so many properties, and all of them are easily remembered and applied in practice. Therefore, the article discusses the basic properties of the degree, as well as where they are applied.
Degree Properties
We consider 12 properties of a degree, including the properties of degrees with the same bases, and we give an example for each property. Each of these properties will help you solve tasks with degrees faster, and also save you from numerous computational errors.
1st property.
and 0 = 1
Many people often forget about this property, make mistakes, representing the number to the zero degree as zero.
2nd property.
a 1 = a
3rd property.
and n * a m = a (n + m)
It must be remembered that this property can be applied only when multiplying numbers; when summed, it does not work! And we must not forget that this, and the following, properties apply only to degrees with the same basis.
4th property.
a n / a m = a (nm)
If in the denominator the number is raised to a negative degree, then when subtracting the degree of the denominator is taken in brackets for the correct replacement of the sign in further calculations.
The property only works when dividing; when subtracting, it does not apply!
5th property.
(a n ) m = a (n * m)
6th property.
a -n = 1 / a n
This property can be applied in the opposite direction. A unit divided by a number to some extent is this number to a minus degree.
7th property.
(a * b) m = a m * b m
This property cannot be applied to sum and difference! When raising a power to a sum or a difference, the formulas of abbreviated multiplication are used, not the properties of the degree.
8th property.
(a / b) n = a n / b n
9th property.
a ½ = √ a
This property works for any fractional degree with a numerator equal to one, the formula will be the same, only the degree of the root will vary depending on the denominator of the degree.
Also, this property is often used in the reverse order. A root of any degree from a number can be represented as a number in the degree unit divided by the degree of the root. This property is very useful in cases where the root of the number is not retrieved.
10th property.
(√ a) 2 = a
This property does not only work with square root and second degree. If the degree of the root and the degree to which this root is raised coincide, then the root expression will be the answer.
11th property.
n √a n = a
You need to be able to see this property in time when solving in order to save yourself from huge calculations.
12th property.
a m / n = n √a m
Each of these properties will be found more than once in tasks, it can be given in its pure form, and it may require some transformations and the use of other formulas. Therefore, for the correct solution, it is not enough to know only properties, you need to practice and connect the remaining mathematical knowledge.
Application of degrees and their properties
They are actively used in algebra and geometry. Degrees in mathematics have a separate, important place. With their help, exponential equations and inequalities are solved, as well as the degrees often complicate equations and examples related to other branches of mathematics. Degrees help to avoid large and long calculations; degrees are easier to reduce and calculate. But to work with large degrees, or with degrees of large numbers, you need to know not only the properties of the degree, but also competently work with the bases, be able to decompose them in order to facilitate your task. For convenience, you should also know the value of numbers raised to a power. This will reduce your time in solving, eliminating the need for long calculations.
The concept of degree plays a special role in the logarithms. Since the logarithm, in essence, is the power of a number.
Abbreviation formulas are another example of using degrees. The properties of degrees cannot be used in them, they are laid out according to special rules, but degrees are always present in each formula of abbreviated multiplication.
Degrees are also actively used in physics and computer science. All transfers to the SI system are carried out using degrees, and later, when solving problems, the properties of the degree are applied. In computer science, degrees of two are actively used, for the convenience of counting and simplifying the perception of numbers. Further calculations of unit conversions or task calculations, just like in physics, occur using degree properties.
Degrees are also very useful in astronomy, where it is rarely possible to find application of the properties of a degree, but the degrees themselves are actively used to reduce the recording of various quantities and distances.
Degrees are used in everyday life, when calculating areas, volumes, distances.
With the help of degrees, very large and very small quantities are recorded in any field of science.
Exponential equations and inequalities
A special place in the properties of a degree is occupied precisely in exponential equations and inequalities. These tasks are very common, both in the school course and in exams. All of them are solved by applying the properties of the degree. The unknown is always in the very degree, therefore, knowing all the properties, it is not difficult to solve such an equation or inequality.