Mathematics is a lesson very close to everyday life. In this lesson, we will discuss things called exponents or power numbers. How did these exponential properties become known?
It is said that this method was first discovered by the mathematician Euclid. He comes from Greece and is also known as the father of geometry. Meanwhile, the first modern use of exponents was made in 1544 by Michael Stifel.
This exponential number is the method chosen by many experts or researchers. Especially when they have to write a large number of zeros or a large number of decimal numbers after zeros.
Understand the foundations
An exponent is a multiplication method that is repeated using the same method. Therefore, this exponent has the form an, where a is the base and n is the power or exponent.
In other words, an exponential number is a number with a power or a number that contains a power. This exponent is a basic mathematical concept that provides a description of the power of a variable or number.
This exponent itself can be a negative number or a decimal number that expresses division with the same power value as the number in question.
Properties of exponents and examples
If you are asked how many exponent properties there are, the answer is that there are many, a total of 8. These properties are usually used to solve various exponent problems. Well, these properties will be explained according to their strength.
In fact, in exponents there are basic laws that apply, namely the law of division and the law of multiplication. However, these two laws are then divided so that the properties of numbers with powers or exponents become as follows.
1. Reduction
The properties of numbers with subtraction powers will be applied to the division formula. The strength of the number will be reduced if the division contains the same base or a. Therefore, the reduction formula will be written like this.
aM : an = amillion
Let’s take an example question as follows:
75 :73 = 75-3= 72 =49
2. Addition
The exponential property of addition is a property that applies to multiplication with the same base or base. In this multiplication calculation, the exponents will be added so that the writing of the formula looks like this.
aM xan = a m + n
Let’s take an example question as follows:
32 × 33 = 32+3 = 35 =243
3. Multiplication
The properties of exponential numbers are multiplication. This property applies to numbers that have powers that are also raised to powers. These exponents will then be multiplied so that the formula looks like this:
(aM)n = aM x N
Let’s take an example question as follows:
(22)2 = 22×2 = 24 = 16
4. Multiplying numbers raised to powers as well
If there is a product raised to a power, this exponential property applies. Later, each number in the multiplication process will be raised to the respective power so the formula is written like this.
(dad)M = aM . BM
Let’s take an example question as follows:
(2 x 4)2 = 22 × 32 = 4 x 14 = 56
5. Arrange the number of fractions
Exponents can also be given on rational numbers. In rational numbers like these, the numerator and denominator must be given powers, but only on the condition that b does not equal 0 or b ≠ 0.
Therefore, the denominator of a number cannot be 0 and to write the exponential properties of this number as follows.
Let’s take an example question as follows:
6. Negative force
This property will be applied to exponential numbers that have negative powers. The way it is calculated is that the power in question is equal to 1 for every power of the exponential number that changes its nature to positive. The formula is as follows:
a – N = 1an.
Let’s take an example question as follows: 4 – 3 = 1.4.3
But there is another explanation as well, where this exponent applies to fractions. If this is the case, if the denominator of the fraction is negative, then when it rises to the top it changes to positive and vice versa. So the formula will be written like this.
1/ An = a-N
For this formula, let’s take an example question like this:
1/35 = 3-5
7. Exponents in fractions
If in this one attribute, for example, there is a number AM And roots with n. If this number is then converted to an exponent, the root of n will change to the denominator while the power of m will change to the numerator.
But provided that the value of n is greater than or equal to 2. The formula is like this.
n√AM = afrom
For this formula, let’s take an example question like this:
4√36 = 36/4
8. Zero energy
As to the nature of this matter, Provision A will apply.0 = 1. If any number is given to the power of 0, the result will still be 1. Take this example, 50this can also be written as follows:
51-1 = 51 × 5-1 = 51 Eleventh/51 = 51/51 = 1
But this also takes into account that the value of a cannot be the same as 0. The reason is that if a = 0 then it will become 00 The result becomes indefinite. Even when calculated using a calculator, there will be no results.
Therefore, you should understand the explanation and examples of the above mentioned properties of foundations properly. Do not confuse the two, because sometimes in one problem, several exponential properties apply simultaneously. If you don’t understand this, you will be confused about solving the problem.
Apply the properties of exponents in everyday life
The properties of exponents are actually formulas that are very close to everyday life, although not everyone feels them. These qualities in themselves are very useful, especially in solving problems in various fields.
Therefore, this material is very important to study. In fact, the use of these properties can be found in the field of biology, in the field of economics and in the social field. This is explained as follows:
1. In biology
In biology, properties of exponential numbers are often used to calculate bacterial growth. For this field, let’s take the following example:
The number of amoeba can increase in a given time because the amoeba will grow by division. The exponential formula used in this case is AR = a0 S (2)R. a0 = 40 at 09.00 am. The question is, how many amoebas were there at 09.08 then?
Solution :
a0 = number of amoeba and t = duration of observation
aR = a0 S (2)R = 100 x (2)8 = 100 x 256
aR = 25.600
This means that in just 8 minutes the number of amoebas can reach 25,600
2. In economics
Aside from biology, foundations are commonly applied in economics. In the economic sector, exponential applications are usually applied in the banking and investment sectors. An example of a case like this.
Suppose you want to invest by buying shares in Company A. The stock price is constantly rising by up to 20% each year. If you buy shares worth 1 million IDR, in 5 years your money will be 2,488,320 IDR.
This is according to the information contained in the following table:
| year to | 0 | 1 | 2 | 3 | 4 | 5 |
| Total money | 1,000,000 Indonesian Rupiah | 1,200,000 IDR | 1,440,000 IDR | 1,728,000 IDR | 2,073,600 Indonesian rupiah | Rs 2,488,320 |
It should be noted that the 20% increase is calculated from the total funds received in the last year, not in the first year where the value was only IDR 1 million. This 20% increase will continue continuously.
Unless there is a case where the stock price falls and the profit you get is less than 20%. This type of exponential increase in economics is usually called compound interest.
Knowing how much profit you will actually get from investing also explains why learning the basics is important. The concept of exponential numbers will be very clear by doubling the profits earned every year.
The reason is that, as mentioned earlier, exponential multiplication is a repeated multiplication method and in the above example the profits obtained are also repeated.
3. Social sector
In the social sector, foundational materials are widely used to calculate and estimate population growth over a given period of time. Take a case like this for example.
In 2017, the population of District A was approximately 286,841 people. What will be the population of the region in 2027 if the population growth rate is 2.99%?
To solve this problem, you need to use the usual population growth rate formula
st= s0Hg The explanation is as follows
sR : Estimated population in 2027
s0: The population in 2017 was 286,841 people
R: time period
Y: population growth rate
e: the exponential number, which is 2.71828182
From what is known above, all that remains is to apply the formula so that the solution looks like this:
sR= p0Hg
sR = 286,841 S0.0299 x 10
sR = 286.841 x 1.34850962347291
sR = 386.807
Examples of exponent properties questions
As mentioned earlier, exponent problems usually consist not only of one property, but of several properties simultaneously. This means that you should understand all the rules about the properties of exponential numbers so that it will be easier to do this.
For this exercise, let’s take an example like this.
What is the result of (8a3)2 ÷ 4 A4 =
From this example, the solution can be written as follows:
82 Sa3)2 ÷ 2a4 (The power of 3 is multiplied by the power of 2 so the result is 6 and is written as follows)
64xa6÷4xa4 = 16 A2 (64 is divided directly by 4 because it is similar and produces the number 16, so with regard to the power of 6 it is reduced to the power of 4 in accordance with the provisions of the nature of exponential numbers, that is, if the division is in the form, then the process of calculating the energy is reduced)
In conclusion, exponents are numbers in multiplication form that are the same over and over again. To better understand these exponents, you need to pay attention to the different properties of numbers with exponents or exponents.
Also do more practice questions on properties of exponents to deepen your understanding. Because most of what appears in the problem is a combination of several exponential properties at the same time, not just one property.