Introduction to Exponents : Definition, Laws, and examples

Introduction to Exponents

Introduction to Exponents: Definition, Laws, and examples

We frequently read numbers in terms like hundred, thousands, lakhs, crores, and so on. What are some integers with more digits than humans can read? The mass of the Earth, for example, is 5972190000000000000000000 kg. This cannot be expressed in simple terms.

Exponents are used to pronounce these types of numbers. This article provides a quick overview of exponents, as well as laws, attributes, and examples.

What is an exponent?

A number’s exponent indicates how many times the number has been multiplied by itself. For instance, 3 * 3 * 3 *3 can be expressed as 34, because 3 is multiplied by itself four times. In this case, 3 is referred to as the “base,” while 4 is referred to as the “exponent” or “power.”

 In general, yn denotes that y has been multiplied by itself n times.

In this case, the term yn

y is known as the “base,” and n is known as the “exponent.” yn is written as “y to the power of n” (or) “y raised to n.”

Use an exponent calculator to find the power of whole numbers in a fraction of a second.

Exponent Laws:

The laws are the set of rules that we have to follow during the procedure. If we break the rules, we obtain an irrelevant solution. To obtain the correct and precise solution we have to follow the laws.

  1. Multiplication law:

When the two or more two terms with the same bases multiply each other, we use the exponent law of multiplication “keep the base same and add the power of the numbers”.

Example

Simplify the expression 24 * 29 * 27.

Solution:

Step 1: Write the given expression 

= 24 * 29 * 27

Step 2: Apply the Multiplication law “keep the base same and add the power of the numbers”.

= 24+9+7

= 220

= 1048576

  1. Division law:

When the two or more two terms with the same bases divide each other, we use the exponent law of division “keep the base same and subtract the power from numerator”.

Example

Simplify the expression 712/75.

Solution:

Step 1: Write the given expression 

= 712/75

Step 2: Apply the Division law “keep the base same and subtract the power from numerator “.

= 712 – 5

= 77

= 823543

  1. Negative exponent law:

The negative exponent law is defined as “If the exponent of a number is negative the write 1/ (number with positive exponent)”. We mostly obtain the answer in the positive exponent.

Example

Simplify the expression 2-2.

Solution:

Step 1: Write the given expression 

= 2-2

Step 2: Use the Negative exponent law “Write the number in the denominator of 1 by making the exponent positive”

= 1/ 22

= 1/ 4

= 0.25

  1. Same exponent law:

If the exponent is the same when numbers are multiplied or divided by each other we “Take the exponent as the whole of all terms”. 

Example

Simplify the expression 22 * 52 * 72.

Solution: 

Step 1: Write the given expression 

= 22 * 52 * 72

Step 2: Use the same exponent law

= (2 * 5 * 7)2

= (70)2

= 4900

Example

Simplify the expression 22 /52.

Solution: 

Step 1: Write the given expression 

= 22 /52

Step 2: Use the same exponent law

= (2 /5)2

= (0.4)2

= 0.16

  1. Zero exponent rule

If the exponent of any number is zero, then “it is always equal to 1”. a0 = 1 where a is any real number.

Example

Solve the expression 20 + 53.

Solution:

Step 1: Write the expression 

= 20 + 53

Step 2: Use the zero-exponent law

= 1 + 5 * 5 * 5 as 20 = 1

= 1 + 125

= 126

  1. Power of exponent

When we have a power of exponent expression than “Multiply the exponent with the power”. Let us have a number with exponent a and also have a power b like (xa)b = xab

Example

Solve the expression (22)5.

Solution:

Step 1: Write the expression 

= (22)5

Step 2: Use the Power of the exponent

= (22)5

= 22 * 5

= 210

= 1024

Examples of exponents

We discuss some complicated examples by using the laws of the exponent.

Example 1

Simplify the expression (23 * 3-2)/ 9-4.

Solution:

Step 1: Write the expression 

= (23 * 3-2)/ 9-4

Step 2: Apply the laws of exponents

= (23 * 3-2)/ (32)-4

= (2*2*2/ 32)/ 3-8

= 2 * 2* 2 * 38/32

 = 8 * 38-2

= 8 * 36

= 8 * 729

= 5832

Example 2

Evaluate (x9 * x-2)/ (y5 * x4).

Step 1: Write the expression 

= (x9 * x-2)/ (y5 * x4)

Step 2: Apply the laws of exponents

= (x9 * x-2)/ (y5 * x4)

= (x9 – 2)/ (y5 * x4)

= x7/ (y5 * x4)

= x7 – 4 / y5

= x3 / y5

y= 2x is an exponential equation with variable x. The graph of this equation shows the increasing rate of the function. 

y = 2-x is an exponential equation with an exponent is –x. This exponential equation shows the decreasing rate of the function.

Application:

Exponents are superscript numerals that indicate how many times a number should be multiplied by itself. Understanding scientific sales solve the pH scale or the Richter scale, utilizing scientific notation to represent very large or very small numbers, and taking measurements are some real-world uses.

 Exponents are also used in real life when you calculate the area of any square. Exponents are needed since it is difficult to write the product when a number is repeated several times by itself. For example, it is quite simple to write 10^7 rather than 10 * 10 * 10* 10 * 10 * 10 *10.

Summary:

In this post, we have learned about the exponential, and different laws with explanations and examples. Now you can solve any exponent problem easily. 

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