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# Definition, Examples | Properties of Exponents

## What is an Exponent

In general, the exponent represents the number of times that a number is multiplied by itself, and the exponentiation operation can be used to perform operations like finding roots and logarithms. In mathematics, an exponent is a symbol used to represent the power to which a number or expression is raised.

For example, in the expression 23, the number 3 is the exponent, and it represents the power to which 2 is raised. The expression 23 is read as “2 raised to the power of 3” or simply “2 to the power of 3”.

The result of the expression 23 is 8, so 2 raised to the power of 3 is equal to 8.

or so 23 = 2 × 2 × 2 = 8

Some more examples:

##### Example: 43 = 4 × 4 × 4 = 64
• In words: 43 could be called “4 raised to the power of 3”, “4 to the power of 3” or simply “4 cubed”
##### Example: 72 = 7 × 7 = 49

Exponents are commonly used in mathematical operations and can be used to simplify complex mathematical expressions. Exponents can also be negative, fractional, or even complex numbers, depending on the context in which they are used.

## Properties of Exponents

Exponents are mathematical symbols that indicate a repeated multiplication of the same number. In mathematical notation, the expression “a^n” represents the result of multiplying “a” by itself “n” times. Here are some of the key properties of exponents:

1. Law of product

If “am” and “an” are two exponential expressions with the same base, then the product of the two expressions are given below.

am × an = am + n

For example, (a2) * (a3) = a^(2 + 3) = a5.

2. Law of quotient

If “am” and “an” are two exponential expressions with the same base we subtract the exponents., then the quotient of the two expressions are given below :

am / an = am – n

For example, (a4) / (a2) =a(4-2) = a2.

3. Law of power of a power

If “am” is an exponential expression, then (a)n  = a(m*n).

For example, (a2 )3  = a(2*3) = a6.

4. Law of zero exponent

If “a0” is an exponential expression, then “a0” = 1.

5. Law of negative exponent

If “a-n” is an exponential expression, then “a-n” = 1 / (an).

For example, a-2 = 1 / (a2).

6. Law of a fraction

If “a(m/n)” is an exponential expression, then “a(m/n) = (a)1/n“.

These properties of exponents are useful for simplifying and solving mathematical expressions. Understanding these properties is essential for advanced mathematical concepts such as logarithms and exponential functions.