The nature of operations on exponential numbers with examples of problems and their solutions

August 15, 2023
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The nature of operations on exponential numbers with examples of problems and their solutions – How do mathematical operations work on numbers? Let's look at the discussion together in the article below to better understand it.

The nature of operations on exponential numbers with examples of problems and their solutions

A number with a power is a number that is used as a simplified form of a number where the number has the same multiplication factors.

So for more details, we can see as follows aⁿ = a x a x a x…..x n where aⁿ denotes exponential numbers, then a is the base number and n itself is the exponent.

For example, we can take one example, namely 5x5x5x5x5, we can simplify it to form 5⁵ if read it becomes five to the power of five.

In exponential numbers, there are several types of exponential numbers, namely positive powers, negative powers, zero powers, and fractional powers.

Exponential numbers are repeated multiplication of a number, where numbers can be positive integers, zero, or negative integers. In simple terms, the writing of this type of number is as follows: a

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ⁿ = a x a x a x…..x a

a is called the base or base number, while n is called the exponent or exponent

Exponential numbers describe the simple form of numbers that have the same multiplication factors as 5 x 5 x 5 x 5 x 5. To facilitate and simplify the process, the writing of the example can be 5⁵.

The properties of integer operations with positive powers for any real numbers a and b and integers m and n apply to the following properties of powers.

a^m × aⁿ = a^m+n
a^m/ aⁿ = (a)^{M N}, m > n, and a ≠ 0
(a^m)ⁿ = a^m×n
(a × b)^m = a^mb^m
(a: b)^m = a^m: a^m, b ≠ 0

Example :

Simplify the exponential form below and write the result in positive exponential form!

b³× b² = b³⁺² = b⁵
b⁷: b³ = b⁷/ b³ = b^7-3 = b⁴
(a⁴b²)³ = a^4×3b^2×3 = a¹²b⁶
a² × a⁶ = a²⁺⁶ = a⁸

There are 3 types of exponential numbers that need to be known, including positive exponential numbers, negative exponent numbers, and zero exponential numbers.

Positive Integer Numbers

Operations with positive integer powers have several properties that can be used to make calculations easier. The following are the properties of these number operations:

Multiplication of exponential numbers

In the first property, the multiplication of these numbers can be written by the formula:

a^m x aⁿ = a^m+n

Example problem: Simplify the multiplication form of this exponential number 4² x4⁴

completion: 4²x4⁴ = 4²⁺⁴= 4⁶

Division of numbers

In the second property, the distribution of exponential numbers can be written by the formula:

a^m: aⁿ = a^{M N}

Example problem: Simplify this form of division of numbers: 3⁶: 3⁴

completion: 3⁶: 3⁴ = 3^6-4= 3²

Rank numbers

The third property can be written by the formula (a^m)ⁿ = a^mxn

Example problem: Simplify this exponential form (3²)⁴?

Completion: (3²)⁴ = 3(^2×4) = 3⁸

Multiplication of Equal Power Numbers

In the fourth characteristic, the following formula can be written: a^m x b^m = (a x b)^m

Example problem: Simplify the multiplication form of this exponential number 2³ x 5³?

Completion: 2³ x 5³ = (2 x 5)³ = 10³

Distribution of Equal Rank Numbers

In the fifth property can be written by the formula

Example problem: determine another form of division of numbers raised to the power of 3⁵/4⁵

Completion: 3⁵/4⁵ = (3/4)⁵

Positive integer exponents indicate that the exponent of a number is positive so that the shape becomes as below.

Y^a = Y x Y x Y x Y x ……x Y

Information:

Y is the number base exponential
a is the number of factors or powers

Based on the form that has been written above, there are several forms that can be learned:

Y shape¹ can be written as Y without having to include the rank in the base
Y value⁰ does not always express the result equal to 1 even though Y is a real number. Because when the form is 0⁰, then the result will be uncertain
A shape that isn't as simple as Y^ab requires more special workmanship because it has different working properties

If you find a Y shape^a+b, then you can use other properties to simplify the shape as below.

Y^a+b = Y^ax Y^b

From these forms, you can break down various exponential forms that have a mix of variables and constants such as Y^7x and its kind. In addition to the working properties above, there are several working properties for positive integer powers as below.

Y^m: Yⁿ = Y^{M N}, for values m > n

(Yⁿ)^a = Y^na

(XY)ⁿ = XⁿYⁿ

(X/Y)^m = X^m / Y^m, for the value of Y ≠ 0

Negative Integer Numbers

Negative integer power numbers have different processing properties because numbers with negative powers need to be converted to fractional form as shown below.

For operations with negative integers, the operations are the same as those with positive integers.

If a is a non-zero number (a ≠ 0) with a negative integer power, then a applies^-n = 1/aⁿ

Example problem: Change shape 5^-2 be a positive exponential number

Solution: bearing in mind the nature of numbers with negative integer powers, the answer is

5^-2 = 1/5² = 1/25

So form a number to a positive power of 5^-2 is 1/25

Zero Power Numbers

The third property to be discussed is the number to the power of zero. Numbers raised to the power of zero have their own special properties because zero does not have any complicated processing operations. Here are some properties of numbers to the power of zero.

X⁰= 1

0^N = 0

0⁰ = Undefined

Every number that has a power of zero has a value of 1 but if it is 0 to a power of zero, then the result is undefined so that X⁰ = 1, for all values x ≠ 0

If a is an integer month zero (a ≠ 0), then a applies⁰ = 1

Example problem: calculate the result of the following powers 10⁰? and 100⁰ ?

Completion: by remembering the value of a⁰ = 1, then 10⁰ = 1 and 100⁰ = 1

Fractional Power Numbers

Fractional exponents have different processing properties than positive integer exponents. Some of the special properties possessed by fractional exponential numbers are as follows.

For all values of m and n ≠ 0. If the values of m and n = 0, then the result is undefined and cannot be resolved.

The nature of operations on exponential numbers with examples of problems and their solutions

Examples of Problem Properties of Exponential Number Operations

Problem 1.

What is the multiplication of 3²x 3⁶

To do the problem above, you can use the addition property of numbers whose exponents are positive integers.

X^a. X^b = X^a+b

3²x 3⁶ = 3²⁺⁶= 3⁸

Problem 2.

Determine the multiplication result of:

To do the problem above, you can simplify the form to the simplest form.

Problem 3

Keep it simple!

(5 x 2) ⁴ =
(a² b⁶ c³) ² =

Answer :

( 5 x 2 ) 4 = 104 = 10000
(a² b⁶ c³) ² = a ^2x2 b ^{6 x 2} c ^{3 x 2} = a⁴ b¹² c⁶

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