Starlight Tools

Exponent Calculator with Steps

Enter a base and exponent to calculate xn, or choose an advanced mode to solve for a missing value. See exact forms, decimal approximations, and worked steps.

Answer

Enter a base and exponent to see your answer.
Advanced: find exponent or base

Find exponent uses logarithms with a positive base other than 1 and a positive result. Find base returns all supported real roots; when there are two, the positive root is carried into other modes.

Settings and input help

Use integers, decimals, fractions (3/4), mixed numbers (1 1/2), e-notation (6.022e23), e, or pi (π). Paste a power such as 2^10 or 27^(2/3) into Base and leave Exponent blank. A base of -4 with exponent 2 means (-4)^2; the pasted expression -4^2 follows standard precedence and means -(4^2).

What does xn mean?

Exponents are a compact way to describe repeated multiplication. The expression xn is read “x to the power of n.” For example, 34 means 3 × 3 × 3 × 3 = 81. This calculator can find the result of a power, the exponent that produces a result, or the base that produces a result.

Negative exponents flip a power into a reciprocal, so x-3 = 1 / x3. Fractional exponents connect to roots: x1/2 = sqrt(x) and x3/2 = sqrt(x3). The calculator accepts common fraction input such as 1/2, 3/4, and -2/3 so you do not need to convert them to decimals first.

Watch negative-base notation carefully. In standard order of operations, -42 means -(42) = -16, while (-4)2 = 16. When you type a negative base in this calculator, it is treated as parenthesized. A negative base with a fractional exponent may not have a real-number result, especially when the denominator is even.

How to use the exponent calculator

  1. Enter a base and exponent, then select Calculate.
  2. Read the exact answer and steps; open Details for alternate formats.
  3. For a missing exponent or base, open Advanced, choose a mode, and enter the two known values.
  4. Adjust decimal display in Settings if needed. Exact answers are unaffected.

Privacy: the calculator evaluates your inputs in your browser.

Worked Examples

3^4 = 81

Multiply four factors of 3: 3 × 3 × 3 × 3 = 81.

2^-3 = 1/8

A negative exponent takes the reciprocal: 2^-3 = 1/(2 × 2 × 2) = 1/8 = 0.125.

7^0 = 1

For a nonzero base, the zero exponent rule gives 1. For example, 7^1 / 7^1 = 7^(1−1) = 1.

27^(2/3) = 9

Take the cube root, then square: (∛27)^2 = 3^2 = 9.

10^2.5 = 100√10

Write 2.5 = 5/2. Then 10^(5/2) = √(10^5) = √(10000 × 10) = 100√10 ≈ 316.227766.

(-4)^2 = 16

Parentheses include the minus sign in the base: (-4) × (-4) = 16.

(-4)^(1/2): no real answer

A square root of -4 is not real because every real number squared is nonnegative. In complex numbers the principal square root is 2i; this tool reports real results only.

Domains and common mistakes

Real-number rules for x^n (reduce p/q to lowest terms first)
Base and exponentReal-number result
x > 0; any real nPositive, including fractional and irrational exponents.
x ≠ 0; integer nDefined. Negative n means a reciprocal; n = 0 gives 1.
x = 0; n > 00.
0^0Undefined by this calculator's convention; other contexts may define it as 1.
x = 0; n < 0Undefined: division by zero.
x < 0; n = p/q with odd qReal: take the odd root then raise to p. Example: (-8)^(1/3) = -2.
x < 0; n = p/q with even q, or irrational nNo real value under these rules. Complex-number evaluation is outside this tool.
  • A negative exponent does not make the answer negative: 2^-3 = 1/8, not -8.
  • Parentheses matter: -4^2 = -(4^2) = -16, while (-4)^2 = 16.
  • Powers do not distribute over addition: (2 + 3)^2 = 25, while 2^2 + 3^2 = 13.
  • A rounded decimal is not the same as an exact fraction: use 1/3 for a cube root, not 0.333333.

Practical applications

Bacteria doubling

Starting with 100 bacteria, six doubling periods give 100 × 2^6 = 100 × 64 = 6,400 bacteria. If each period is 20 minutes, that is two hours in an ideal model with no growth limits.

Repeated 5% growth

A quantity that increases by 5% each period is multiplied by 1.05. After 12 periods, 1.05^12 ≈ 1.795856326. Starting with 1,000 units gives about 1,795.856326 units, a total increase of about 79.586%, assuming the same growth rate each period.

Calculation methodology and references

Integer and fraction inputs, including finite decimals, are reduced to exact BigInt ratios before powers are evaluated. Integer powers use exact arithmetic; rational powers first check for perfect roots and simplify small radicals. Larger radicals remain in exact symbolic form. Non-rational powers use exp(n ln x) for positive x. Missing exponents use ln(y)/ln(x); missing bases use reciprocal powers with real-domain checks.

To keep the browser responsive, expanded exact powers are limited to 10,000 digits per numerator or denominator and exponent magnitude 100,000. Perfect-root checks are limited to degree 1,024 and 2,000-digit radicands. Beyond these limits, the exact power is retained symbolically. Input expressions are limited to 12,000 characters, 256 tokens, and 32 nested parentheses.

Decimal approximations use IEEE 754 binary64 arithmetic (53 bits of precision): typically 15–17 significant decimal digits, with only about 15 reliably retained across general calculations. This is not a guarantee of 17 accurate digits. Integers above 9,007,199,254,740,991 are not all exactly representable as JavaScript Numbers; BigInt results avoid that limit. Display rounding (0–12 decimal places) cannot restore lost precision. Overflow above roughly 1.798 × 10^308 and underflow below roughly 4.94 × 10^-324 are reported explicitly.

Exponent Laws

These laws apply to positive bases and real exponents, or to integer exponents wherever defined. Denominators must be nonzero. For negative bases and fractional exponents, check the real domain before combining powers.

RuleFormulaExample
Product rulex^a * x^b = x^(a+b)2^3 * 2^4 = 2^7 = 128
Quotient rulex^a / x^b = x^(a-b)5^6 / 5^2 = 5^4 = 625
Power rule(x^a)^b = x^(a*b)(3^2)^4 = 3^8 = 6561
Power of a product(xy)^a = x^a y^a(2*5)^3 = 2^3 * 5^3 = 1000
Power of a quotient(x/y)^a = x^a / y^a(6/2)^3 = 6^3 / 2^3 = 27
Negative exponentx^-a = 1 / x^a2^-4 = 1 / 16 = 0.0625
Zero exponentx^0 = 1, x != 08^0 = 1
Exponent of 1x^1 = x12^1 = 12
Fractional exponentx^(a/b) = b-root(x^a)27^(2/3) = cuberoot(27^2) = 9

Exponent FAQ

What is an exponent?

An exponent tells how many times to use the base as a factor. For example, 4^3 = 4 × 4 × 4 = 64.

How do negative exponents work?

They make a reciprocal: x^(-n) = 1/x^n, as long as x is not zero.

What is x^0?

For any nonzero x, x^0 = 1. This follows from the quotient rule: x^a / x^a = x^0 = 1.

What is 0^0?

It depends on context, but this calculator marks 0^0 as undefined to avoid mixing conventions.

How do fractional exponents work?

For positive x, x^(1/2) is √x, and x^(3/2) is (√x)^3. In general, x^(p/q) = (q-th root of x)^p. For negative x, reduce p/q first; a real result requires odd q.

How do you multiply or divide exponents?

With the same base, add exponents when multiplying and subtract exponents when dividing: x^a × x^b = x^(a+b), and x^a / x^b = x^(a-b). Division requires a nonzero base; use these laws only where the powers are defined.

Why does -4^2 differ from (-4)^2?

Without parentheses, the exponent applies before the negative sign: -4^2 = -16. With parentheses, the negative base is squared: (-4)^2 = 16.

How do you find an unknown exponent?

For a positive base b not equal to 1 and positive result y, solve b^n = y with n = ln(y) / ln(b).

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