We often read numbers in words such as hundred, thousands, lakhs, crores and so on. What numbers have more digits than we can read? For example, the mass of Earth is 5972190000000000000000000 kg. This cannot be read in simple words. Thus, to pronounce these types of numbers we make use of exponents. In this article, a brief introduction of exponents is given along with rules, properties and examples.
Exponent Meaning
Exponent is defined as the method of expressing large numbers in terms of powers. That means, exponent refers to how many times a number multiplied by itself. For example 6 is multiplied by itself 4 times, i.e. 6 × 6 × 6 × 6. This can be written as 6^{4}. Here, 4 is the exponent and 6 is the base. This can be read as 6 is raised to the power 4.
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Exponent Symbol
The symbol used for representing the exponent is ^. This symbol (^) is called a carrot. For example 4 raised to 2 can be written as 4^2 or 4^{2}. Thus, 4^2 = 4 × 4 = 16.
Exponent Laws
Different laws of exponents are described based on the powers they bear.
Multiplication Law: Bases – multiplying the like ones; add the exponents and keep the base the same.
When bases are raised with power to another, multiply the exponents and keep the base the same.
Division Law: Bases – dividing the like ones; subtract the exponent of the denominator from the exponent of the numerator Exponent and keep the base the same.
Let ‘a’ be any integer or a decimal number and ‘m’, ‘n’ are positive integers, that represent the powers to the bases such that the above laws can be written as:
- a^{m }. a^{n }= a^{m+n}
- (a^{m})^{n} = a^{mn}
- (ab)^{n} = a^{n} b^{n}
- (a/b)^{n} = a^{n}/b^{n}
- a^{m}/a^{n} = a^{m-n}
- a^{m}/a^{n} = 1/a^{n-m}
These laws referred to the properties of exponents. These are used to simplify complex algebraic expressions and write the large numbers in an understandable manner.
Exponent and Powers
As defined above, exponent defines the number of times a number multiplied by itself. The power is an expression that shows repeated multiplication of the same number or factor. For example, in the expression 6^{4}, 4 is the exponent and 6^{4} is called the 6 power of 4. That means, 6 is multiplied by itself 4 times.
Learn more about exponents and powers here.
Exponent Formula and Rules
Exponents have certain rules which we apply in solving many problems in maths. Some of the exponent rules are given below.
Zero rule: Any number with an exponent zero is equal to 1.
Example: 8^{0} = 1, a^{0} = 1
One Rule: Any number or variable that has the exponent of 1 is equal to the number or variable itself.
Example: a1 =a, 71 = 1
Negative Exponent Rule: If the exponent value is an negative integer, then we can write the number as:
a^{-k} = 1/a^{k}
Example: 3^{-2} = 1/3^{2} = 1/(3 × 3) = 1/9
Exponent Table
The below table shows the values of different expressions in terms of exponents along with their expansions and values. This will help you in understanding the simplification of numbers with exponents in detail.
Type of Exponent |
Expression |
Expansion |
Simplified value |
Zero exponent |
6^{0} |
1 |
1 |
One exponent |
4^{1} |
4 |
4 |
Exponent and power |
2^{3} |
2 × 2 × 2 |
8 |
Negative exponent |
5^{-3} |
1/5^{3} = 1/(5 × 5 × 5) |
1/125 |
Rational exponent |
9^{1/2} |
√9 |
3 |
Multiplication |
3^{2} × 3^{3} |
3^{(2 + 3)} = 3^{5} |
273 |
Quotient |
7^{5}/ 7^{3} |
7^{(5 – 3)} = 7^{2} |
49 |
Power of exponent |
(8^{2})^{2} |
8^{(2 × 2)} = 8^{4} |
4096 |
Example
Example: Simplify (3^{2} × 3^{-5})/ 9^{-2}
Solution:
(3^{2} × 3^{-5})/ 9^{-2} = 3^{(2 – 5)} × 9^{2}
= 3^{-3} × (3^{2})^{2}
= 3^{-2} × 34
= 3^{(-2 + 4)}
= 3^{2}
= 9
Therefore, Simplify (3^{2} × 3^{-5})/ 9^{-2} = 9
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