Exponent Rules You Actually Need
Every exponent problem reduces to a handful of rules. Master these and you can simplify any expression without a calculator.
Product Rule: a^m × a^n = a^(m+n)
When you multiply two powers with the same base, add the exponents. 2³ × 2&sup4; = 2&sup7; = 128. You're counting the total number of times the base is multiplied by itself: (2 × 2 × 2) × (2 × 2 × 2 × 2) = 2&sup7;.
Quotient Rule: a^m ÷ a^n = a^(m−n)
When you divide two powers with the same base, subtract the exponents. 5&sup6; ÷ 5² = 5&sup4; = 625. Cancel out the common factors in the numerator and denominator.
Power of a Power: (a^m)^n = a^(m×n)
When you raise a power to another power, multiply the exponents. (3²)³ = 3&sup6; = 729. You're applying the exponent operation n times, each time multiplying the exponent by m.
Root Properties
Roots are the inverse of exponents. The nth root of x asks “what number, raised to the nth power, gives x?”
√(a × b) = √a × √b — The root of a product equals the product of the roots. √(36) = √(4 × 9) = 2 × 3 = 6.
√(a / b) = √a / √b — The root of a quotient equals the quotient of the roots. √(25/4) = 5/2 = 2.5.
The nth root of a = a^(1/n) — Any root can be written as a fractional exponent. √8 = 8^(1/2), ∛27 = 27^(1/3) = 3. This lets you apply exponent rules to roots.
Logarithm Laws
Logarithms answer the question “what exponent do I need?” log₂(8) = 3 because 2³ = 8. Three laws handle every simplification.
Product law: log(a × b) = log(a) + log(b) — Logs turn multiplication into addition. log₁₀(200) = log₁₀(2 × 100) = log₁₀(2) + 2 = 2.301.
Quotient law: log(a / b) = log(a) − log(b) — Logs turn division into subtraction. log₁₀(50) = log₁₀(100/2) = 2 − 0.301 = 1.699.
Power law: log(a^n) = n × log(a) — The exponent comes down as a multiplier. log₁₀(1000) = log₁₀(10³) = 3 × log₁₀(10) = 3. This is how you solve for unknown exponents in equations like 2^x = 1024.
Common Powers of 2
Powers of 2 appear constantly in computing (memory sizes, binary, data storage) and in everyday math. Here are the ones worth knowing by heart.
| Expression | Value | Context |
|---|---|---|
| 2¹ | 2 | Binary digit |
| 2² | 4 | Nibble (half byte) |
| 2³ | 8 | 1 byte = 8 bits |
| 2&sup4; | 16 | Hexadecimal base |
| 2&sup5; | 32 | 32-bit color depth |
| 2&sup6; | 64 | 64-bit architecture |
| 2&sup7; | 128 | ASCII character set size |
| 2&sup8; | 256 | RGB color per channel |
| 2&sup9; | 512 | Common SSD size (GB) |
| 2¹&sup0; | 1,024 | 1 KB (kibibyte) |
| 2¹¹ | 2,048 | Common RSA key size |
| 2¹² | 4,096 | 4K page size |
| 2¹³ | 8,192 | Audio buffer size |
| 2¹&sup4; | 16,384 | 16K texture resolution |
| 2¹&sup5; | 32,768 | Max signed 16-bit integer |
| 2¹&sup6; | 65,536 | Max unsigned 16-bit integer |
For working with very large or very small numbers, the scientific notation calculator converts between standard and exponential form. For operations with fractional exponents and mixed numbers, the fraction calculator breaks down each step.