Exponents

An Introduction to Exponents

When it is 100, 29, or even 135000, it is easier to read these numbers as a whole. However, consider the number 294800000000, was this simple to read? Maybe yes, but somehow takes time to evaluate. But then, when the same number is denoted as 2948 × 108, it was quick and easy to say out. Now, this method of using powers to express a long set of natural numbers is called Exponents. 

This module is focussed on giving you an introduction to exponents, along with a few examples and types. 


Defining What Exponents Mean

The terms Exponent, Index, and Power all mean the same. When an expression or statement of specific natural numbers (0 to ∞), are represented as repeated power by multiplication of its units, then the resulting number is an Exponent. The resultant set of numbers is the same as that of the original sequence. 

Let us understand this concept with a simple example number. 

Take the value 84. Here the number 8 is called “base” and the number 4 (up) is called the exponent or power of that mathematical sequence. Now, to accurately calculate the value of this exponent, simply multiply the base number as many times as denoted by the power. So for this case, it would be 8 × 8 × 8 × 8 = 4096. Hence, 4096 can be represented in the form of exponents as 84


Four Major Forms of Exponents

In the process of getting an introduction to exponents, we will now learn the 4 major types of Indices, subjected upon the value present as its power:

  1. Rational exponent - Square or Cube roots turn radical. The number is simplified by having the denominator of the exponent outside the root and keeping the base number as root, with its power as the numerator. 

  2. Positive exponent - A number is simplified by multiplying its base with the number of times mentioned in its power. 

  3. Negative exponent - The value is estimated by using 1 in its numerator and base, plus the exponent in its denominator.

  4. Zero exponent  - The set does not even have to be calculated since any exponent with the value of 0 is equal to 1. 


Simple Examples to Understand Exponents

  • Base 10 and power 3 is denoted as 103. Now, you can find its value by multiplying 10 (base) 3 times (power digit), which is 10 × 10 × 10 = 1000.

  • The number 23450000000 can be exponentially denoted as 2345 × 107.

  • With an exponent value of 4 and the base as 2, i.e. 23, the natural number is obtained by multiplying 2 three times. Hence, the answer is 2 × 2 × 2 = 8.

  • The base 1000 with its Index value 1 is represented as 10001. The simplification of this exponent is 1000 since the power value is only 1 and the base value remains unchanged. 

  • 716929 × 103 can be numerically expressed as 716929000.

Make sure to check the sign of both the base and exponent, as 2 negative signs will give you a positive value. And the odd count of negative exponents will give you a negative result. Again, consider the instance -103. Now the answer is -10 × -10 × -10 = -1000 and not simply 1000. However, in the case of -102, the answer is -10 × -10 = 100 (because - × - gives +).


A Gist about Negative Numbers with Examples

Similar to positive exponents, a negative number is equated by having the reciprocal of the positive value obtained from an expression. Say, that the base 6 has the negative power value of -2. This is symbolized as 6-2

Now, calculate the digit’s value by moving the negative exponent at the denominator, i.e. reciprocal of the positive power. Hence it becomes 1/62 which is 1/36.

Here are 2 more examples of negative exponents.

  1. 4-3: 1/43 which is 1/64.

  2. 10-5: 1/105 upon simplification gives 100000.


Conclusion

Exponent or index represents the power of units present in a number sequence. Exponents can be observed in 4 different types namely, positive, negative, zero and rational/fractional. The number’s value can be interpreted by using the exponent as the total number of times the base number has to be multiplied with the same base.  For negative powered values, the positive exponent’s reciprocation gives the value of the number and the result will be in the fractional form.

Even though there are multiple exponent values possible, the powers 1 and 0 will result in the same numbers which are 1 and 0 respectively. No 2 values are noted together with an exponent digit simultaneously.

FAQ (Frequently Asked Questions)

1. Can a Number Have its Negative Exponent as 0?

No. A number can never have 0 as its negative exponent since the resulting answer will again be 0 as the denominator is also changed as 0 (anything divided by 0 is 0).

2. What is the Value of a Number with its Exponent Position as Simply 0?

The value of a number with its exponent position having simply 0 equals 1. This is common to any natural number regardless of its length.

3. Give The Value Of The Exponential Form 00.

00 is simplified and evaluated as 1. Any base number having its index value as 0 will be equal to 1 only. This question is also called the “Zero Exponent Rule”.

4. 23 Or 32. Which Exponent Of The 2 Has The Highest Value To The Other?

Evaluate both the exponent by using the simplification method. 23 is 2 × 2 × 2 =8 and the value of 32 is 3 × 3 = 9. Since 8<9, the exponent 32 possesses a higher value than 23.

5. As Per Mathematical Operations, What Is The Difference Between An Exponent And A Power?

Technical speaking, both exponents and powers define the same meaning. Yet, both the terms are used interchangeably. The Power is a number used to represent the total count of repeated multiplication for any given number. On the other hand, an exponent is the power’s value that has to be raised for that given natural number.

6. State 1 Real-life Example For Exponents.

Exponents are used in the field of graphic designing, to evaluate and judge the size and volume of a product designed online. The graphic creator must identify the area of a particular shape using its appropriate formula and for this, exponents can help in detecting the size of a figure with high accuracy.