Exponents
How about calculating the distance between moon and earth? You will be surprised that the distance is even greater than lakhs. The actual distance between moon and earth is 3, 84,400 Km. But it is really difficult to remember such a bigger distance value.
So, what can we do? We can try to simplify this bigger value. Why not write 3,84,400 as 6202. 620* 620 = 3, 84,400.
Thus, the exponent of a number implies about the number of times to use the number in a multiplication.
What are Exponents?
A simple natural number when expressed in the form of power, then it is called an exponent or power or indices.
In other words, we can also say that exponents are expressed in the form of multiple of the simplest numbers. Thus, by the exponent, you can simplify or reduce a bigger number into a much smaller number.
Exponents Examples
For Example, 625 can be written as 252 or 25 * 25 = 625. Here, 25 is the base and 2 is the exponent. Exponents can also be termed as power. And thus you can also say 2 is both exponent and power.
Exponent is extremely useful in regular mathematics. When you have large digits, then it is not possible to remember them fully. And thus they are expressed in the form of exponents.
For Example, 5612161 is such a large number. You can simplify and write it as, 2369 * 2369 = 5612161.
You can also write it as, 23692 = 5612161.
Types of Exponents
There are several types of exponents on the basis of their power. Let’s have a look at some of the major types of exponents:
Positive Exponent – These exponents are the ones which have positive numbers as their power. For Example, 49 = 72 is a positive exponent.
Negative Exponent – These exponents are the ones which have negative numbers as their power. For Example, 8-2. This needs to be simplified.
Let’s understand how to simplify a negative exponent.
When you have the number 8-2.
You need to first add 1 in the numerator and convert the number in question into the denominator. For Example, 8-2 = -1/8 * 1/8 = 1/64.
Thus, 8-2 can also be written as 1/64.
Zero Exponent – These exponents are the ones when you have 0 in the power. 0 is equal to 1 and thus you don’t need to perform any special operation here.
For Example, 90 is equal to 91. It can also be written as,
90 = 91 = 9.
Rational Exponent – These exponents are the ones which have a rational number or fractions as the power. For Example, 101/2 is the example of a rational exponent.
101/2 can be called as 2 roots of 10.
Rule of Exponents
Following are the simple but extremely useful rules of exponents:
A0 = 1: As per this rule, the power of a natural number is 0, and then the result will be 1.
AmAn = Am+n: As per this rule, when there are two same natural nos., then the different powers can be added together.
Am/n = Am-n: As per this rule, when power is in the rational form, then the numerator and denominator can be subtracted and brought to simple terms.
(Am)n = Am.n: As per this rule, when there are brackets between two different powers then it can be multiplied and then solved accordingly.
The above rules will help in solving any of the mathematical equations.
For Example,
(52)3 = 52*3 = 56 = 5 * 5 * 5 * 5 * 5 * 5 = 15625
Let’s solve one more example,
Simplify, 642/22
= (64/2)2
= (32)2
= 1024
Relation between Positive and Negative Power
ax = 1/a-x
Where, a is the base. X is the power or exponent.
Let’s understand this with an example,
52 = 1/5-2
Solved Examples:
Example: Simplify 62 + 43 + 72 - 23
Solution:
= 6 * 6 + 4 * 4 * 4 + 7 * 7 – 2 * 2 * 2
= 36 + 64 + 49 – 8
= 149 – 8
= 141
Example: Solve 16/22 + 32/42
Solution:
= 16/2*2 + 32/4*4
= 16/4 + 32/16
= 4 + 2
= 6
Example: Solve 102 * 103 + 25
Solution:
= 100 * 1000 + 32
= 100000 + 32
= 100032
Example: Solve 172 * 183
Solution:
= 289 * 5832
= 1685448
Fun Facts
Exponents, on contrary to multiplication, do NOT "distribute" over addition.
Anything raised to the power zero (0) is just "1" (until and unless "anything" is not itself zero).
FAQs on An Introduction to Exponents
1. What are exponents and powers in Maths?
An exponent is a mathematical notation that indicates how many times a number, called the base, is multiplied by itself. For example, in the expression 5³, the number 5 is the base and 3 is the exponent or power. It means you multiply 5 by itself 3 times: 5 × 5 × 5 = 125. This provides a shorthand way to write repeated multiplication.
2. What are the main laws of exponents?
The main laws of exponents help simplify expressions. For any non-zero integers 'a' and 'b' and any integers 'm' and 'n', the key laws are:
Product Law: aᵐ × aⁿ = aᵐ⁺ⁿ
Quotient Law: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Power of a Power Law: (aᵐ)ⁿ = aᵐⁿ
Power of a Product Law: (ab)ᵐ = aᵐbᵐ
Power of a Quotient Law: (a/b)ᵐ = aᵐ/bᵐ
Zero Exponent Law: a⁰ = 1 (where a ≠ 0)
3. What do the zero and one exponents mean?
The rules for exponents of zero and one are fundamental and have specific outcomes:
Exponent of One: Any number raised to the power of 1 is the number itself. For example, 12¹ = 12.
Exponent of Zero: Any non-zero number raised to the power of 0 is always equal to 1. For example, 7⁰ = 1.
4. Why does any non-zero number raised to the power of zero equal one?
This concept can be understood using the quotient law of exponents, which states that aᵐ ÷ aⁿ = aᵐ⁻ⁿ. If we take a case where m = n (and 'a' is not zero), such as a³ ÷ a³, the law gives us a³⁻³ = a⁰. At the same time, we know that any number divided by itself equals 1. Since both a⁰ and 1 are the results of the same calculation (a³ ÷ a³), it logically follows that a⁰ = 1.
5. What is the difference between expressions like (–3)⁴ and –3⁴?
The placement of parentheses is critical because it determines what the exponent applies to. This is a common point of confusion.
In (–3)⁴, the base is '–3'. The exponent 4 applies to the entire negative number, meaning (–3) × (–3) × (–3) × (–3), which results in a positive 81.
In –3⁴, the base is just '3'. The exponent 4 applies only to the base '3', and the negative sign is applied after. This means –(3 × 3 × 3 × 3), which results in a negative –81.
6. How are exponents used to express very large numbers in science?
Exponents are essential for writing very large or small numbers in a compact form called scientific notation (or standard form), as per the CBSE syllabus. A number is expressed as a product of a decimal number between 1.0 and 10.0 and a power of 10. For instance, the distance from Earth to the Sun is about 149,600,000,000 metres. In scientific notation, this is written as 1.496 × 10¹¹ m, making it much easier to read and use in calculations.
7. In what real-world scenarios are exponents applied?
Exponents are not just for classroom maths; they appear in many real-world applications:
Finance: To calculate compound interest, where the investment grows exponentially.
Biology: To model the exponential growth of bacteria in a petri dish.
Computer Science: To measure memory and storage capacity, where kilobytes, megabytes, and gigabytes are based on powers of 2.
Physics: To describe phenomena like radioactive decay or the Richter scale for earthquakes, which is logarithmic (the inverse of exponential).
