Powers & Exponents Calculator
Calculate powers, or solve for an unknown base or exponent, with step-by-step work.
Powers & Exponents Tool
Calculate powers, find unknown bases or exponents with detailed explanations and step-by-step solutions
Quick Examples
Powers and exponents
an means multiply a by itself n times: 23 = 2 × 2 × 2 = 8. Negative exponents flip to a reciprocal (2-3 = 1/8). Fractional exponents are roots (41/2 = 2). Any non-zero a0 = 1.
Watch order of operations with negatives: (-2)4 = 16, but -24 = -16.
Quick reference
- Positive integer: repeated multiplication
- Zero: a0 = 1 (a ≠ 0)
- Negative: a-n = 1/an
- Fractional: a1/n = ⁿ√a
Exponent rules
Same base only. Different bases do not combine this way.
(a + b)n is not an + bn.
Logs are the inverse: if ax = b, then x = loga(b). For large powers by hand, use repeated squaring (binary exponentiation). Crypto and modular arithmetic take the modulo at each step so values stay manageable.
Where exponents show up
- Physics: inverse-square laws, radioactive decay, decibels
- Finance: compound interest A = P(1 + r/n)nt; present value uses negative exponents
- CS: complexity like O(2n), RSA modular exponentiation, binary tree height
- Biology: growth N(t) = N₀ert, drug half-life
Common mistakes
- Treating (a + b)n as an + bn
- Mixing am × an with (am)n
- Confusing a-n with -an
- Forgetting a0 = 1
- Apply product/power rules only with matching bases
- Use parentheses with negative bases
- Check by expanding small cases or estimating
Frequently Asked Questions
Why is a0 = 1?
From the quotient rule: an ÷ an = a0, and any non-zero number divided by itself is 1.
(-2)4 vs -24?
Parentheses raise the signed number: (-2)4 = 16. Without them, the negation applies after the power: -24 = -16.
How do I compute large powers by hand?
Break the exponent into powers of two. For 313, use 13 = 8+4+1 so 313 = 38 × 34 × 3.
What is a fractional exponent?
a1/n is the nth root; am/n = (ⁿ√a)m. Example: 81/3 = 2.
How do exponents relate to logs?
They are inverses. To solve for a variable in the exponent, take a log of both sides.
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