Exponent Calculator
Calculate base raised to a power (exponentiation). Type 'e' as the base for Euler's number.
The Power of Exponents: Growing Fast and Shrinking Fast
Exponents are a shorthand way to show repeated multiplication. Instead of writing 2 times 2 times 2 times 2 times 2, you write 2 to the 5th power, which equals 32. This simple notation unlocks some of the most powerful ideas in mathematics. Exponents describe how populations grow, how radioactive materials decay, how light dims over distance, and how your money grows with compound interest. They are one of the most useful and widely applicable tools in all of math.
Our calculator handles every type of exponent problem: positive exponents, negative exponents, zero exponents, fractional exponents, and large exponents. Enter the base and the exponent and get the result instantly, along with a step-by-step explanation of the calculation.
The basic idea: Base to the power of exponent means multiply the base by itself exponent times. Two to the 3rd power = 2 times 2 times 2 = 8. Five to the 4th power = 5 times 5 times 5 times 5 = 625. Ten to the 6th power = 1,000,000 (one million). The base is the number being multiplied, and the exponent tells you how many times to multiply it.
The Rules of Exponents
Product rule: When you multiply the same base with different exponents, add the exponents. Two to the 3rd times two to the 4th = two to the 7th = 128. Think of it this way: three factors of 2 times four factors of 2 equals seven factors of 2 total. This rule turns multiplication problems into simple addition, which is much easier to work with.
Quotient rule: When you divide the same base with different exponents, subtract the exponents. Two to the 7th divided by two to the 3rd = two to the 4th = 16. You are canceling out three of the seven factors of 2, leaving four. The power rule: (two to the 3rd) to the 4th = two to the 12th = 4,096. An exponent raised to another exponent means multiply the exponents.
Zero and negative exponents: Any number to the zero power equals 1. Five to the 0 = 1, 100 to the 0 = 1, even 0 to the 0 is defined as 1 in most contexts. A negative exponent means reciprocal: two to the negative 3rd = 1 / (two to the 3rd) = 1/8. Negative exponents flip the number upside down, turning large numbers into tiny fractions and tiny fractions into large numbers.
Scientific Notation: Handling Enormous and Tiny Numbers
What is scientific notation? It is a way to write very large or very small numbers using exponents. A number in scientific notation looks like a x 10 to the n, where a is between 1 and 10, and n is an integer. The speed of light is about 3 x 10 to the 8th meters per second (300,000,000 m/s). The width of a human hair is about 7 x 10 to the negative 5th meters (0.00007 m).
Why scientists love it: The distance to the nearest star is about 4 x 10 to the 16th meters. Writing that out as 40,000,000,000,000,000 meters is error-prone and hard to read. Scientific notation makes it compact, clear, and easy to compare. Multiplying and dividing these numbers is simple too: multiply the coefficients and add (or subtract) the exponents.
Everyday examples: The population of Earth is about 8 x 10 to the 9th (8 billion). A gigabyte of data is about 10 to the 9th bytes. The age of the universe is about 1.38 x 10 to the 10th years. A single grain of sand weighs about 1 x 10 to the negative 5th kilograms. Once you learn to read scientific notation, you gain the ability to comprehend numbers that would otherwise be just a confusing string of zeros.
Exponents in Real Life
Compound interest: Money grows exponentially. If you invest $1,000 at 7% annual interest compounded yearly, after 30 years you have $1,000 times 1.07 to the 30th = about $7,612. Your money more than triples without you adding another cent. This is the power of exponential growth, and it is the reason financial advisors say the most important factor in building wealth is time, not the amount you invest.
Bacterial growth: A single bacterium that divides every 20 minutes becomes 2 to the 72nd (about 4.7 sextillion) bacteria after just 24 hours. Of course, real bacteria eventually run out of food and space, but the early exponential growth phase is incredibly fast. This is why food left at room temperature can become dangerously contaminated in just a few hours, and why understanding exponential growth is crucial for food safety.
Computing and data: Computer memory sizes double repeatedly: kilobyte (2 to the 10th), megabyte (2 to the 20th), gigabyte (2 to the 30th), terabyte (2 to the 40th). Each step up is about a thousand times larger. Moore's Law observed that the number of transistors on a computer chip doubles roughly every two years, an exponential growth trend that has driven the entire digital revolution for over five decades.
Fractional Exponents and Roots
What is a fractional exponent? An exponent of 1/2 means square root. An exponent of 1/3 means cube root. In general, an exponent of 1/n means the nth root. So 9 to the 1/2 = 3 (the square root of 9), and 27 to the 1/3 = 3 (the cube root of 27). Fractional exponents and roots are two ways of writing the same thing, and the exponent form is often more convenient for calculations.
Mixed exponents: An exponent like 5/2 means take the square root and then raise to the 5th power. So 16 to the 5/2 = (16 to the 1/2) to the 5th = 4 to the 5th = 1,024. You can do the root first or the power first, whichever is easier. This flexibility makes fractional exponents a powerful tool for simplifying complicated expressions.
Why roots matter: Square roots tell you the side length of a square from its area. Cube roots tell you the edge length of a cube from its volume. If a square garden has an area of 64 square feet, the side length is the square root of 64 = 8 feet. If a cube-shaped box has a volume of 125 cubic inches, each side is the cube root of 125 = 5 inches. Roots (fractional exponents) are the key to going backwards from area and volume back to side lengths.
Frequently Asked Questions
An exponent tells you how many times to multiply a number by itself. It is written as a small number above and to the right of the base number. For example, 2 to the 3rd power (2 cubed) means 2 x 2 x 2 = 8. The base is the number being multiplied, and the exponent is how many times you multiply it. It is a shortcut for writing repeated multiplication.
A negative exponent means the reciprocal, or one divided by the number. So 2 to the power of -3 equals 1 divided by 2 to the 3rd power = 1/8 = 0.125. Think of it as the number moving to the denominator. This is incredibly useful in science and engineering because it lets you write very small numbers compactly.
Any nonzero number to the power of zero equals 1. This is a fundamental rule of math: 5 to the 0 power = 1, 100 to the 0 power = 1, even 999,999 to the 0 power = 1. It seems strange at first, but it makes perfect sense because each time you divide the exponent by 1, the result should shrink proportionally, and 1 is where that pattern naturally leads.
When multiplying numbers with the same base, add the exponents. 2 cubed times 2 squared = 2 to the 5th power. When dividing, subtract the exponents. 2 to the 5th power divided by 2 squared = 2 to the 3rd power. When raising a power to another power, multiply the exponents. (2 cubed) squared = 2 to the 6th power. These rules make working with exponents much faster.
Exponents describe exponential growth and decay, which appear everywhere. Population growth, compound interest, radioactive decay, spread of viruses, and computer processing power all follow exponential patterns. Without exponents, we could not express very large numbers (like the distance to stars) or very small numbers (like the size of atoms) efficiently.
Scientific notation writes very large or very small numbers using exponents of 10. The speed of light is about 3 x 10 to the 8th power meters per second (300,000,000 m/s). The size of an atom is about 1 x 10 to the power of -10 meters (0.0000000001 m). Exponents make these extreme numbers readable and easy to work with.
2 squared (2 to the power of 2) means 2 x 2 = 4, and geometrically it represents the area of a square with sides of length 2. 2 cubed (2 to the power of 3) means 2 x 2 x 2 = 8, and it represents the volume of a cube with sides of length 2. The names come from the 2D square and 3D cube shapes they represent.
Try More SupaCalc Tools
Free calculators for finance, health, AI costs, and more.
Browse All Calculators