Exponents in Mathematics: Definitions, Rules & Examples

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 it takes time to evaluate. But then, when the same number is denoted as \[2948 \times 10^{8}\], it was quick and easy to say. Now, this method of using powers to express a long set of natural numbers is called Exponents. 

This module is focused 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 \[\infty\]), 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 \[8 \times 8 \times 8 \times 8 = 4096\]. Hence, 4096 can be represented in the form of exponents as \[8^{4}\].

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 \[10 \times 10 \times 10 = 1000.\]

• The number 23450000000 can be exponentially denoted as \[2345 \times 10^{7}.\]

• With an exponent value of 4 and the base as 2, i.e. \[2^{3}\], the natural number is obtained by multiplying 2 three times. Hence, the answer is \[2 \times 2 \times 2 = 8.\]

• The base 1000 with its Index value 1 is represented as \[1000^{1}\]. The simplification of this exponent is 1000 since the power value is only 1 and the base value remains unchanged. 

• \[716929 \times 10^{3}\]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 \[-10^{3}\]. Now the answer is\[ -10 \times -10 \times -10 = -1000\] and not simply 1000. However, in the case of \[-10^{2}\], the answer is \[-10 \times -10 = 100 (because - \times - gives +).\]

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 \[\frac{1}{6^{2}}\] which is \[\frac{1}{36.}\]

Here are 2 more examples of negative exponents.

1. \[4^{-3} : \frac{1}{4^{3}} \]which is \[\frac{1}{64}\].

2. \[10^{-5} : \frac{1}{10^{5}} \] upon simplification gives 100000.

Rules of Exponents

Exponent characteristics, often known as exponent laws, are used to solve issues involving exponents. These characteristics are also known as major exponents rules, which must be obeyed while dealing with exponents. The rules are simple and can be remembered with a little practice. Exponents are added, subtracted, multiplied, and divided according to some of the most prevalent principles. It's vital to keep in mind that these rules only apply to real numbers.

Zero Exponent Property- According to this characteristic, every integer raised to the power of zero equals one. For example, \[2^{0} = 1.\]

Negative Exponent Property - This property implies that simply flipping the fraction, any negative exponent may be transformed to a positive. The number, however, must not be 0.  For example, 2^-3 would be written and solved as \[\frac{1}{2^{3}} = \frac{1}{8}\]. 

Products of Power Property - This condition indicates that when multiplying the same number by different exponents, the exponents can be added together. The number can't be 0. For example, \[2^5 \times 2^3 = 2^ {(5+3)} = 2^8 = 256.\]

Quotient of Powers Property - When dividing the same number with distinct exponents, this rule specifies that the exponents must be subtracted. The integer can't be 0. For example, \[\frac{2^{5}}{2^{3}} = 2^ {(5-3)} = 2^{2} = 4.\]

Power of a Product Property - When two or more distinct numbers with the same exponent are multiplied, the exponent is only utilised once, according to this characteristic.

Quotient of a Product property - This property indicates that dividing two separate numbers with the same exponent is solved by dividing the integers first, then applying the exponent. For example, \[\frac{4^{3}} { 2^{3}} = (\frac{4}{2}) ^{3} = 2^{3} = 8.\]

Power to a Power Rule - This rule indicates that when a power is raised to another power, the exponents are multiplied. For example, \[(2^{3})^{2} = 2^{(3 \times{2})} = 2^{6} = 64.\]

Negative Exponents

Negative exponents show that the power of a number is negative, and they also apply to its reciprocal. An exponent is the number of times a number is multiplied by itself, as we all know. A negative exponent is the multiplicative inverse of the base raised to a power that is exactly opposite to the power provided. In other words, we write the number's reciprocal and then solve it like positive exponents. A negative exponent indicates how many times the reciprocal of the base must be multiplied.

For example, if it is given that a-n, it can be expanded as \[\frac{1}{an}\]. It means we have to multiply the reciprocal of a, i.e \[\frac{1}{a}\] 'n' times. Negative exponents are used while writing fractions with exponents.

Fractional Exponents

A fractional exponent is a number's exponent that is a fraction. Powers and roots can be represented using fractional exponents. A is the base, and b is the exponent, in any generic exponential equation of the type ab. A fractional exponent is defined as the value of b expressed in fractional form. Square roots, cube roots, and the nth root are all fractional exponents.

Few examples of fractional exponents are \[\frac{21}{2}, \frac{32}{3}\], etc. The general form of a fractional exponent is\[\frac{\times {m}}{n}\], where \[\times\] is the base and \[\frac{m}{n}\] are the exponent.

Importance of Exponents

Exponents are significant since it is difficult to express products when a number is repeated several times without them. These are useful in mathematics because they allow us to compress information that would otherwise be extremely long to write. If we wanted to describe the product of \[\times\] multiplied by itself 7 times in mathematics, we'd only be able to write it as \[\times\times\times\times\times\times\times\], \[\times\] multiplied by itself 7 times in a row if we didn't use exponents. So, we need a different way to describe that value, which is where exponents come in. Instead of writing out x multiplied by itself 7 times, we can write \[\times^7\].

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.