What Are Exponents?
Exponents are a way to represent repeated multiplication of a number by itself. The number being multiplied is called the base, and the number of times it is multiplied is called the exponent or power.
Example: In , the base is 2, and the exponent is 3. This means:
Parts of an Exponential Expression
- Base: The number that is being multiplied repeatedly.
- Exponent: The small number written above and to the right of the base, indicating how many times the base is multiplied.
Example:
In 5⁴, 5 is the base, and 4 is the exponent. It means 5 x 5 x 5 x 5 .
Laws of Exponents
Zero Exponent Rule
Any base raised to the exponent of 0 is always equal to 1 (except for 0 itself).a0 = 1
Example: 70 = 1
Product of Powers Rule
When multiplying two expressions with the same base, you add the exponents.am x an = am+n
Example: 32 x 35 = 32+3 = 35 = 243
Power of a Power Rule
When raising a power to another power, you multiply the exponents.(aᵐ)n = amxn
Example: (23)2 = 23x2 = 26 = 64
Power of a Product Rule
When a product of two or more factors is raised to a power, each factor is raised to that power individually and then multiplied.(ab)m = amxbm
Example: (3x4)2 = 32x42 = 9x16 = 144
Identity Exponent Rule
Any number raised to the exponent of 1 is equal to the number itself.a1 = a
Example: 101 = 10
Quotient of Powers Rule
When dividing two expressions with the same base, you subtract the exponents.am ÷ an = am-n
Example: 54 ÷ 52 = 54-2 = 52 = 25
Negative Exponent Rule
A negative exponent means the reciprocal of the base raised to the positive exponent.a⁻ᵐ = $\frac{1}{aᵐ}$
Example: 4-2 = $\frac{1}{4²}$ = $\frac{1}{16}$
Power of a Quotient Rule
When a quotient (a fraction) is raised to a power, both the numerator and the denominator are raised to that power individually. ᵐ($\frac{a}{b}$)m = $\frac{aᵐ}{bᵐ}$
Example: ($\frac{2}{5}$)³ = $\frac{2³}{5³}$ = $\frac{8}{125}$
Why Are Exponents Important?
Exponents are a powerful mathematical tool used in many areas, such as:
- Science: To represent large numbers (e.g., the distance between stars) or very small numbers (e.g., the size of atoms).
- Finance: In compound interest calculations, exponential growth models represent how money grows over time.
- Computer Science: Exponential functions are used in algorithms, data structures, and computing power analysis.
Examples of Exponents in Real Life
- Square and Cube of Numbers:
When you square a number, you multiply it by itself once, and when you cube it, you multiply it by itself twice.- Squaring:
- Cubing:
- Growth Models:
Population growth and bacterial growth often follow exponential models. If a population doubles every year, it follows an exponential pattern.