Exponents Studio
Power up (literally). Calculate exponents, explore the rules of growth, and reference Powers of 2 & 10.
Power Calculator
Calculate exponents instantly. Visualize negative powers and the logic of repeated multiplication.
Step-by-Step
The Power of Powers
Addition is slow. Multiplication is faster. But Exponents are the rocket fuel of mathematics. They describe everything from the explosion of bacteria to the compounding of your savings account.
Efficiency
Scientists hate writing zeros. Instead of 1,000,000,000, they write 10⁹. Exponents are the ultimate compression tool for big numbers.
Compound Growth
"The most powerful force in the universe is compound interest." - Einstein (maybe). Exponents let us predict future value: A = P(1+r)^t.
The Small Scale
Negative exponents let us dive into the microscopic. A nanometer is 10⁻⁹ meters. They are the language of atoms and wavelengths.
The 5 Laws of Exponents
Memorize these patterns, and you can simplify complex algebraic messes in seconds. (Check the **Laws Sandbox** tab to see them in action!)
- 1. Product of Powers:
x^a · x^b = x^(a+b)
Ex: 2³ · 2² = 2⁵ - 2. Quotient of Powers:
x^a / x^b = x^(a-b)
Ex: 2⁵ / 2² = 2³ - 3. Power of a Power:
(x^a)^b = x^(a·b)
Ex: (2³)² = 2⁶ - 4. Zero Exponent:
x^0 = 1
Ex: 9999^0 = 1 - 5. Negative Exponent:
x^-a = 1 / x^a
Ex: 2^-1 = 0.5
Frequently Asked Questions
What is an Exponent?
An exponent (or power) tells you how many times to multiply a number (the base) by itself. In 5³, 5 is the base multplied by itself 3 times: 5 × 5 × 5.
Why is x^0 always equal to 1?
Think of it as dividing. 2³/2³ = 8/8 = 1. But by exponent rules, 2³/2³ = 2^(3-3) = 2^0. Therefore, 2^0 must equal 1.
What does a negative exponent mean?
A negative exponent means taking the reciprocal (flipping the fraction). x^-n is simply 1 / x^n. It represents repeated division instead of multiplication.
How do fractional exponents work?
They represent roots. x^(1/2) is the Square Root of x. x^(1/3) is the Cube Root. x^(2/3) means squaring x, then taking the cube root.
What is the Product Rule?
When multiplying terms with the same base, you add the exponents. 2³ × 2⁴ = 2⁷.
What is the Quotient Rule?
When dividing terms with the same base, you subtract the exponents. 2⁵ / 2² = 2³.
What is Scientific Notation?
It's a way to write very large or small numbers using powers of 10. For example, 3,000,000 is written as 3 × 10⁶.
Why are powers of 2 so fundamental?
Computers run on binary (0s and 1s). Memory sizes like 16GB, 32GB, 256GB are all powers of 2.
What is Exponential Growth?
It's when something grows by multiplication rather than addition. It starts slow but explodes rapidly (like a viral video or compound interest).
Is 0^0 defined?
In most algebraic contexts, it is considered 1. However, in calculus limits, it is an indeterminate form.