Exponent Laws and Rules
All exponent manipulation follows eight core rules. Mastering these rules lets you simplify any exponential expression without a calculator.
Product Rule
aแต ร aโฟ = a^(m+n)
2ยณ ร 2โด = 2^7 = 128
Quotient Rule
aแต / aโฟ = a^(m-n)
2โถ / 2โด = 2ยฒ = 4
Power of a Power
(aแต)โฟ = a^(mรn)
(2ยณ)โด = 2^12 = 4096
Power of a Product
(ab)โฟ = aโฟ ร bโฟ
(2ร3)โด = 2โด ร 3โด = 16 ร 81 = 1296
Power of a Quotient
(a/b)โฟ = aโฟ / bโฟ
(2/5)ยณ = 8/125 = 0.064
Negative Exponent
aโปโฟ = 1 / aโฟ
2โปยณ = 1/8 = 0.125
Zero Exponent
aโฐ = 1 (a โ 0)
7โฐ = 1 ยท 1000โฐ = 1 ยท ฯโฐ = 1
Fractional Exponent
a^(m/n) = โฟโ(aแต)
8^(2/3) = โ(8ยฒ) = โ64 = 4
โน๏ธ
The zero exponent rule follows directly from the quotient rule. Since aโฟ / aโฟ = a^(n-n) = aโฐ, and any nonzero number divided by itself equals 1, aโฐ must equal 1. The case 0โฐ is indeterminate in analysis but is conventionally defined as 1 in combinatorics and computing.
Negative Exponents
A negative exponent does not make the result negative. It means take the reciprocal of the base raised to the positive exponent. For any nonzero base a:
a^(-n) = 1 / a^n
Example: 5^(-2) = 1 / 5ยฒ = 1/25 = 0.04
Example: (1/3)^(-2) = 1 / (1/3)ยฒ = 1 / (1/9) = 9
The second example shows an important pattern: a fraction raised to a negative exponent flips and becomes the reciprocal fraction raised to the positive exponent. (a/b)^(-n) = (b/a)^n.
Negative Exponents in Scientific Notation
Negative powers of 10 represent small numbers. 10^(-3) = 0.001, 10^(-6) = 0.000001 (one millionth). Scientific notation uses this to express tiny quantities: the diameter of a hydrogen atom is approximately 1.06 ร 10^(-10) metres. The negative exponent shifts the decimal point to the left.
| Expression | Result | As Fraction |
|---|
| 10^(-1) | 0.1 | 1/10 |
| 10^(-3) | 0.001 | 1/1,000 |
| 2^(-4) | 0.0625 | 1/16 |
| 5^(-2) | 0.04 | 1/25 |
| (1/2)^(-3) | 8 | 8/1 |
Fractional Exponents and Roots
A fractional exponent represents a root operation. The denominator of the exponent fraction is the index of the root; the numerator is the power applied to the radicand.
a^(1/n) = โฟโa (the nth root of a)
a^(m/n) = โฟโ(aแต) = (โฟโa)แต
Both forms give the same result. The second form often uses smaller numbers.
Common Fractional Exponents
| Exponent | Root Equivalent | Example |
|---|
| 1/2 | Square root | 9^(1/2) = 3 |
| 1/3 | Cube root | 27^(1/3) = 3 |
| 1/4 | 4th root | 81^(1/4) = 3 |
| 2/3 | Cube root, then square | 8^(2/3) = 4 |
| 3/2 | Square root, then cube | 4^(3/2) = 8 |
โ ๏ธ
Fractional exponents with negative bases can produce imaginary numbers. (-8)^(1/3) = -2 is valid (real cube root), but (-4)^(1/2) = 2i is imaginary because no real number squared equals -4. This calculator returns NaN for inputs that would require imaginary numbers.
The Number e and Natural Exponentiation
Euler's number e โ 2.71828182845904523536 is one of the most important constants in mathematics. It is the unique positive real number whose exponential function (e^x) is its own derivative. This property makes e the natural base for growth and decay processes.
e = lim(nโโ) (1 + 1/n)^n โ 2.71828182845905
Also: e = 1 + 1/1! + 1/2! + 1/3! + 1/4! + โฆ (infinite series)
Key Values of e^x
| x | e^x | Common Use |
|---|
| -โ | 0 | Limit of decay |
| -1 | 0.367879441 | Time constant in RC circuits |
| 0 | 1 | Any base to the zero power equals 1 |
| 1 | 2.718281828 | Definition of e |
| 2 | 7.389056099 | Compound interest with continuous compounding |
| ln(2) โ 0.693 | 2 | Doubling time formula |
| ฯ โ 3.14159 | 23.14069263 | Gelfond's constant |
| iฯ | -1 | Euler's identity: e^(iฯ) + 1 = 0 |
Continuous Compound Interest
The formula A = Pe^(rt) gives the amount after time t for principal P at continuous annual interest rate r. For $10,000 at 5% for 10 years: A = 10,000 ร e^(0.05ร10) = 10,000 ร e^0.5 = 10,000 ร 1.6487 = $16,487.21. Compare with annual compounding: $10,000 ร (1.05)^10 = $16,288.95. Continuous compounding adds $198.26 over 10 years.
Frequently Asked Questions
What is an exponent?
An exponent tells you how many times to multiply a base number by itself. In 2^5 (read "2 to the fifth power"), 2 is the base and 5 is the exponent. The result is 2 ร 2 ร 2 ร 2 ร 2 = 32. The exponent can be any real number: a positive integer means repeated multiplication, a negative exponent means the reciprocal, a fractional exponent means a root, and zero always gives 1 (for any nonzero base).
How do you solve for the base when you know the exponent and result?
Take the nth root of the result, where n is the exponent. Base = Result^(1/exponent). For example, if base^3 = 125, then base = 125^(1/3) = 5. For base^2 = 49, base = 49^(1/2) = 7. If the exponent is even and the result is positive, there are two solutions: a positive base and a negative base (since (-7)ยฒ = 49 as well). This calculator returns the principal (positive) root.
How do you solve for the exponent when you know the base and result?
Use the logarithm: exponent = log(result) / log(base) = log_base(result). For example, if 2^n = 64, then n = log(64) / log(2) = logโ(64) = 6. For 3^n = 81: n = log(81) / log(3) = logโ(81) = 4. This works for any base and any positive result. The calculator uses the change-of-base formula automatically when you select the Solve Exponent mode.
What is a negative base raised to a power?
For integer exponents: a negative base raised to an even integer gives a positive result ((-3)ยฒ = 9), and raised to an odd integer gives a negative result ((-3)ยณ = -27). For fractional exponents, results can be imaginary if the denominator of the fraction is even - for example, (-4)^(1/2) = 2i (imaginary). This calculator returns NaN for inputs that would produce imaginary numbers.
What does x^0 equal?
Any nonzero number raised to the power of 0 equals 1. This follows from the quotient rule: x^n / x^n = x^(n-n) = x^0, and any nonzero number divided by itself equals 1. So x^0 = 1 for all x โ 0. The case 0^0 is technically indeterminate (the limit depends on how you approach it), but in combinatorics, computing, and many practical contexts it is defined as 1.
