Exponents
What are Exponents?
An exponent is a number that indicates how many times a base number is multiplied by itself. It is written as a small number (the exponent) to the upper right of the base number.
Exponent Notation
b = base
The number that is being multiplied by itself.
n = exponent (or power)
The number of times the base is multiplied by itself.
Example
Here, 2 is the base and 3 is the exponent. The expression means 2 multiplied by itself 3 times.
Interactive Exponent Calculator
Use this interactive tool to calculate powers and see how exponents work.
Base (b)
Exponent (n)
Exponent Rules
Understanding the rules of exponents helps simplify complex expressions and solve problems more efficiently.
Product Rule
When multiplying powers with the same base, add the exponents.
8 × 16 = 128
Quotient Rule
When dividing powers with the same base, subtract the exponents.
32 ÷ 4 = 8
Power Rule
When raising a power to another power, multiply the exponents.
82 = 64
Zero Exponent Rule
Any number (except 0) raised to the power of 0 equals 1.
This is true for any non-zero base.
Negative Exponent Rule
A negative exponent means to take the reciprocal of the base raised to the positive exponent.
The reciprocal of 8 is 1/8.
Applications of Exponents
Scientific Notation
Scientific notation uses exponents to express very large or very small numbers in a more manageable form.
Large Numbers
Speed of light in meters per second
Small Numbers
Mass of a proton in kilograms
Compound Interest
Exponents are used in finance to calculate compound interest, which is interest calculated on the initial principal and also on the accumulated interest.
Where A is the final amount, P is the principal, r is the interest rate, and t is the time period.
Growth and Decay
Exponential functions model many natural phenomena, such as population growth, radioactive decay, and the spread of diseases.
Where N(t) is the quantity at time t, N0 is the initial quantity, k is the growth/decay rate, and e is the mathematical constant (approximately 2.71828).
Related Topics
Practice Exponents
Test your knowledge with our interactive exponent practice exercises.
Start PracticeKey Concepts
- 1An exponent represents repeated multiplication of the same number.
- 2The product rule states that when multiplying powers with the same base, add the exponents.
- 3The quotient rule states that when dividing powers with the same base, subtract the exponents.
- 4Any non-zero number raised to the power of zero equals 1.
- 5A negative exponent means to take the reciprocal of the base raised to the positive exponent.