In studying mathematics, **rank number** be one of the chapters to be explored. This is because exponential numbers are basic knowledge which will later be useful for later chapters in arithmetic activities.

From solving equations, to helping to calculate large numbers more quickly, exponents are indispensable. That’s why this is a mandatory discussion contained in the lesson chapter at school.

Meanwhile, this number which is often also referred to as an exponent has several inherent properties that you should know about. Keep reading this explanation to find out about these characteristics along with examples of questions and answers!

**What’s that ****Rank Number**** ****or Exponent****?**

Simply put, the number of rank or **exponential is** multiplication between the same number according to the sum of the powers. For example, the number 2 is multiplied by the number 2 4 times, to produce 2x2x2x2, then it can be changed to 2 to the power of 4 (2^{4}).

From there, the general form of rank numbers is as follows:

a^{n}= a x a x a x a x…x a

**example**:

- 5
^{3 }= 5 x 5 x 5 = 125 - 2
^{5 }= 2 x 2 x 2 x 2 x 2 = 32

However, please note that because there are properties in exponential numbers, the general form of exponential numbers above only applies to positive exponents.

Meanwhile, exponential numbers have such a big role in mathematics. Just imagine, if a scientist carried out a study and calculated the speed of light with a result of 300,000,000.

Of course, this number looks very impractical and makes you even more dizzy. To overcome this, the numbers above can be abbreviated by using exponential numbers. So that result can be written as 3 x 10^{8}.

Then, what is often asked is, what if exponents are in the form of fractions, roots, or negative values? This will be summarized in the properties possessed by **rank number**. To find out more, see the explanation below.

**The General Formula for Exponential Numbers**

The general formula for exponential numbers or numbers is **general form** from the promoted number. In general, the form of the rank number is as follows:

a^{b}provided that:

- a is not number one. Because regardless of the power of a number, if the number itself is worth one then the result will still be 1.
- b is a member of the real numbers. For example, powers are included in the numbers 2,4,5, -3, -2, and so on.

Meanwhile, if described, in the formula above,* a* referred to as **basis** (basic number of trees), temporary *b* referred to as **group** (exponent).

**Anything ****Type ****Number of Ranks? **

If you follow the exponential sign, there are four types of exponential numbers, each of which has different properties and formulas. Here are the four types of exponential numbers along with examples that you can learn.

**1. Number of Positive Ranks**

This type of exponential numbers are numbers that have positive exponents. For clarity, here’s the general formula for positive exponential numbers:

Information:* a* is the basis (number of trees) and *n *is rank.

From the formula above it can be explained that the base *a* which is promoted with *n* produce *a* which is multiplied by *a,* so on until the number is the same as the rank.

**Example**:

From the example above, it can be concluded that when a number has a higher power, the value of that number will also automatically increase.

**2. Negative Power Numbers**

After knowing the first type, of course by knowing the name of the second type, you definitely already know. Negative exponents have a meaning that reverses from the first point. In this type, the exponential of a number is a negative number.

In general, negative exponential numbers are formulated as follows:

a^{-n} = ( 1a)^{m}

Information: *a *is a real number and not a number 0, and *m *is a positive integer.

**Example**:

Apart from having an inverse meaning, the nature of this second point is also inversely proportional to the nature of point 1. In negative power numbers, the greater the negative power, the smaller the value of the number.

**3. Zero Power Numbers**

The next type is when a number has a power of zero, then no matter how many numbers there are, the result is 1. Here’s the general formula for numbers to the power of zero:

a^{0 }= 1

Information: *a *are real numbers and *a *is not the number 0.

**Example**:

**4. Number of Fractional Ranks**

For numbers that have exponents in the form of fractions, there are several ways to solve them which can be seen later in the discussion of properties **rank number**. Meanwhile, the general formula for this type of exponential number is as follows:

a^{1/m }= p which is a positive real number, then, p^{m} = a

a^{m/n }= (a^{1/n})^{m}

Information: *a *is a real number and not 0, and *m *is a positive integer.

**Example: **

8^{1/3 }= 2, for 2^{3} = 8

2^{2/3} = (2^{1/3})^{2}

**Properties of Power Numbers**

In meng**operate numbers** exponents or exponents, you can’t just go around and have a good understanding of the properties of this number. This is very important because these properties will later become the main reference in operating exponential numbers.

**1. Nature of Rank Addition**

When there is a multiplication between two or more rank numbers with the same base (prime number), then the property of rank summation will be applied. That is, this property of one applies to the multiplication of numbers with the same base.

When numbers of ranks with the same base are multiplied, then add the ranks.

a^{b }x a^{c} = a^{b+c}

**Example:**

- 2
^{3}x 2^{2}= 2^{3+2 }= 2^{5} - 2
^{3}x 2^{4}x 2^{2}= 2^{3+4+2}= 2^{9}

From the example above, it can be observed that the nature of adding powers can be done because the base or base numbers are the same, namely in the form of the number 2. Meanwhile, if the bases are not the same, then the nature of adding powers cannot apply.

**2. Nature of Downgrade**

Almost the same as the previous point, the basic concept of the nature of subtraction will apply if the base or base numbers are the same. The difference is that if you multiplied it before, this time the power reduction property applies to numbers with the same base being divided.

So it can be concluded, if the number of ranks with the same base is divided, then reduce the rank.

Similar to the previous point, this rank reduction property will not occur if the number of trees or bases is not the same.

**3. Nature of Multiplication of Ranks**

The multiplication property of exponents can be applied to **rank number** which **promoted** again. That is, if there is a number raised to a power, then raised to another power, then multiply the power.

(a^{b})^{c} = a^{b }^{x c}

**Example:**

- (5
^{2})^{3}= 5^{2 x 3 }= 5^{5} - (2
^{3})^{2}= 2^{3 x 2 }= 2^{6}

This property of power multiplication also applies to power numbers multiplied by the same power number, where the base and power are the same.

**Example:**

So the question is, what if the base is the same, but the rank is different? So you can’t apply this property, and can solve it by using the addition property of powers.

**4. Characteristics of Rank Distribution **

The division property of exponents will apply to the radical model. So, before going any further about the nature of division of powers, it would be better to study the shape changes that occur in the form of root numbers to exponents first.

When a number is in the root, then when changed in the exponent will be **rank number** fraction.

**Formula**:

After understanding this concept, the nature of the distribution of ranks will be easier to digest. This property can be applied to exponential numbers**will**. Please be careful, because what is divided later is the rank, not the base. For simplicity, here’s an example:

**5. First Class Traits**

The number 1 is an iconic number in mathematical arithmetic operations. This is because multiplication or division by one will produce the number itself. Of course, the same thing applies to ranks.

If you think about it, exponents are the simplest form of base multiplication raised to the power. That is, if 2^{3 }then it means that 2 multiplied by two is 3 times (2 x 2 x 2). So what if 2 is raised to the power of 1, it means that there is only the number 2 itself without needing to be multiplied again

From this it can be concluded that no matter how many bases there are, if raised to the power of 1 it will produce the base number itself.

a^{1 }= a

**Example: **

- 2
^{1 }= 2^{ } - 25
^{1 }= 25 - 789
^{1}= 789 - 179862
^{1}= 179862

**Negative Traits**

One of the conditions that must be met by a power is to be a part of a real number. If we discussed earlier positive exponential numbers, don’t forget that negative exponential numbers are also possible.

So, what are the properties of negative exponential numbers? Check out the general formula below:

a^{-b }= 1/a^{b}

From the shape of the numbers above, it can be concluded that negative exponents will form a fraction. In this case the quantifier is 1 and the denominator is the base with the power itself.

**Problems example ****Rank Number**** and Answer**

Because it has many properties, its use is also very diverse in solving problems.

To further deepen your understanding of exponential numbers, here are some questions and answers that you can observe and learn from!

**1. Example Question 1**

Set value *y *which can satisfy equation 6^{2y-4 }= 36^{3}

**Discussion:**

To find the value of y, the method is to use the substitution model with the first step equalizing the numbers on the left and right sides. Then think of the exact base, absolutely correct, the answer is 6 because 36 is the product of 6^{2.}

After that, make 36 into 6 squared so that the powers of the two can be equalized so that this problem can be solved, thus:

So, the value of y that can satisfy the equation is 5.

**2. Sample Question 2**

What is the result of the arithmetic operation (-5)^{3 }+ (-5)^{2} + (-5)^{1} + 5^{?
}

**Discussion:**

To solve the problem above, it will be easy if you solve the questions one by one, so:

- (-5)
^{3 }= (-5) x (-5) x (-5) = -125 - (-5)
^{2 }= (-5) x (-5) = 25 - (-5)
^{1 }= -5 - 5
^{0 }= 1

-125 + 25 + (-5) + 1 = -104

So, the result of the arithmetic operation above is -104.

After knowing in full, starting from the definition, types, characteristics, to examples of **rank number**, surely learning Mathematics for the future will be even more exciting. Keep up the enthusiasm and don’t forget to memorize the properties of the exponential numbers above, OK?

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Originally posted 2022-12-03 09:26:42.