Exponent Calculator — Powers, Roots & Big-Number Results

Keep exploring

Three companion tools for powers and logs.

How it works

Two numbers in, the whole family of answers out.

The math behind the power

Go deeper

Decoding negative and fractional exponents — and taming huge answers.

What is an exponent?

One small raised number that replaces a long multiplication.

An exponent is a counting instruction. In 3⁴, the large number — 3 — is the base, the number being multiplied, and the small raised 4 is the exponent, telling you how many times the base appears in the multiplication: 3 × 3 × 3 × 3 = 81. That is the whole definition. Everything else on this page — roots, reciprocals, growth curves — is what happens when you let that counter be something other than a whole number, and the exponent calculator above exists because the counting gets tedious long before it gets difficult: nobody wants to multiply 2 by itself twenty times to learn that the answer is 1,048,576.

The one reading habit worth fixing early is word order, because English names the two numbers in the opposite order to the way they are written. “3 to the power of 4” puts the base first and lands on 81; “4 to the power of 3” swaps them and lands on 64 — a different question with a different answer, 4 × 4 × 4. When someone says “the power of” a number, the number after “of” is the exponent; when a result is described as “a power of 2”, it means the result is what 2 produces at some exponent — 8 is a power of 2 because it is 2 to the power of 3. Keep base and exponent in their places and every rule further down this page is bookkeeping.

Any base, any power: how the calculator handles them

Whole numbers, negatives, fractions and decimals — and solving backwards.

Most tools that call themselves a power calculator quietly assume you want a whole-number exponent and a positive base. This one makes no such assumption, and it is worth knowing what each kind of input means before you lean on the answer. A whole-number exponent is the plain counting case: 3 to the power of 5 is 3 × 3 × 3 × 3 × 3 = 243. A negative exponent returns the reciprocal — 2⁻³ is 1 ÷ 2³ = 0.125 — never a negative result. A fractional exponent is a root: 9^0.5 is √9 = 3. And a decimal exponent is the fractional idea typed for a keyboard: 2^0.1 = 1.071773, a number no counting argument produces, reached by the same smooth curve that passes through every whole power of 2 either side of it.

The calculator also runs in the two backward directions, because “what is the result?” is only one of the three questions powers ask. Solving for the exponent — 2 raised to what gives 32? — is a logarithm in disguise: log(32) ÷ log(2) = 5. Solving for the base — what cubed gives 27? — is a root: 27^(1 ÷ 3) = 3. Pick the missing piece in the “What are you solving for?” menu and the same engine works the problem from the other end, showing its check line (for example, 2⁵ = 32) so you can see the answer land.

Three inputs get an honest refusal instead of a faked number, and meeting them here first saves confusion later. 0⁰ is undefined — the “power 0 gives 1” rule and the “0 to a power is 0” rule collide on it, and mathematics leaves it unsettled. 0 to a negative power would mean dividing by 0. And a negative base with a fractional exponent, like (−2)^0.5, has no real-number answer at all — the square root of a negative is not hiding somewhere on the number line. Everything else, from 10⁶ = 1,000,000 down to 5⁻² = 0.04, is fair game.

Powers, answered: the tables

The highest-demand values, printed — no click required.

Most visits to an exponent page are not really asking for a lesson; they are asking for one number. The questions typed into search boxes are strikingly specific: 3 to the power of 4 is 81. 4 to the power of 3 is 64 — the mirror image, and a smaller number, because a bigger base raised fewer times can lose to a smaller base raised more. 2 to the power of 3 is 8, 2 to the power of 4 is 16, 3 to the power of 5 is 243, 3 to the power of 2 is 9, 5 to the power of 4 is 625 and 6 to the power of 4 is 1,296. The three tables below gather the families those questions belong to, computed once and printed in full, so the answer is on the page rather than behind a button.

Squares: 1² to 30²

Squares from 1² to 30². The diagonal of this table — 5 squared is 25, 6 squared is 36 — is the set of answers most worth knowing by sight.

Cubes: 1³ to 15³

Cubes from 1³ to 15³. Note 4 to the power of 3 = 64 sitting alongside 2 to the power of 6: the same value reached by two different powers.

Powers of 2: 2⁰ to 2²⁰

Powers of 2 from 2⁰ to 2²⁰. This is the computing ladder: 2¹⁰ = 1,024 is the “kilo” of data sizes, and 2²⁰ is the “mega” rung, 1,048,576.

Two patterns in these tables repay a second look. First, the squares grow by odd numbers: each square exceeds the one before by 3, 5, 7, 9… — because (n + 1)² − n² = 2n + 1, the staircase of squares is built from the odd numbers in order. The most familiar square of all sits in the middle of that staircase: 5 to the power of 2 is 25. Second, the powers of 2 double relentlessly: adding 1 to the exponent is the cheapest operation in mathematics and the most violent in effect, which is why 2 to the power of 7 is only 128 while 2 to the power of 20, thirteen doublings later, is past a million. And the two ladders secretly meet: 4 to the power of 5 is 1,024 — exactly the value of 2 to the power of 10 — because 4 is itself 2², so five multiplications by 4 are ten multiplications by 2 with the factors paired up. For values beyond these tables — powers of 10, of 3, of anything — the calculator at the top is the table, continued on demand.

Negative and fractional exponents

The two extensions that turn a counting trick into a system.

Neither extension was invented for elegance; both are forced by keeping one pattern unbroken. Walk the powers of 2 downward — 2³ = 8, 2² = 4, 2¹ = 2 — and each step down divides by 2. Continue the staircase honestly and 2⁰ must be 1, 2⁻¹ must be ½, 2⁻² a quarter: a negative exponent means “1 divided by the positive power”, so 2⁻¹⁰ = 1 ÷ 1,024 = 0.000977, which is exactly the reciprocal row the calculator prints. This is the arithmetic a negative exponents calculator performs, and the subtlety worth keeping is that the minus sign lives in the exponent, not the answer: 2⁻³ is a small positive fraction, 0.125, not a negative number. Only when the base itself is below 1 does the flip push the value upward — (1/2)⁻³ = 8, because flipping a small fraction makes a large whole.

The fractional extension is forced by the product rule instead. If 9^(1/2) is to obey “add exponents to multiply”, then 9^(1/2) × 9^(1/2) must equal 9¹ = 9 — and the only number that multiplies by itself to give 9 is 3. So a fractional exponent is a root: the denominator names which root, the numerator is an ordinary power applied before or after. 8^(2/3) is the cube root of 8, squared: 2² = 4. 25^(3/2) is the square root of 25, cubed: 5³ = 125. And 81^(1/4) is the fourth root of 81: 3, since 3⁴ = 81. Any fraction exponent calculator — this one included — is doing precisely that translation under the hood, which is why typing the exponent as a decimal (0.6667 for 2/3) gets you approximately the answer while typing the fraction’s meaning gets you the exact one.

The two extensions in one view: a minus sign flips the value into a reciprocal, a fraction turns it into a root. Every value computed by the same engine as the calculator above.

Two cautions complete the picture, and both are places where an exponent fraction calculator has to be honest rather than helpful-looking. First, the order inside a mixed exponent like 8^(2/3) does not matter — root-then-square and square-then-root give the same 4 — but square-then-root on a negative base breaks: (−8)^(2/3) computed as “the cube root of −8, squared” is 4, while “the square of −8, cube-rooted” tempts a route through positive 64 that lands in the same place only by luck of this example; the real-valued answer exists here because the root is odd. Second, a fractional exponent on a negative base with an even root — (−4)^(1/2) — has no real answer at all, and the calculator says so rather than inventing one. Fractions in the exponent are roots; roots of negatives are where the real numbers end.

Exponential expressions and functions

When the variable climbs into the exponent, everything changes speed.

Every power so far has had a fixed exponent. An exponential expression lets the exponent vary: 2^x is not one number but a whole family of them, one for every x, and written as f(x) = 2^x it becomes an exponential function — the subject of a different, larger search that an exponential function calculator or a general exponential calculator addresses by graphing and solving rather than by single evaluations. The distinction that matters is which slot holds the variable. In x² the variable is the base and the growth is polynomial: doubling x multiplies the result by 4. In 2^x the variable is the exponent and the growth is explosive: increasing x by 1 doubles the result, whatever x already is. Same symbols, opposite machinery.

The doubling family is the honest way to feel it. Start with a single grain and double the pile thirty times — once per day for a month — and the count runs 1, 2, 4, …, through 2¹⁰ = 1,024 at day ten and 2²⁰ = 1,048,576 at day twenty, finishing at 2³⁰ = 1,073,741,824: over a billion grains, with half of them arriving on the final day. That back-loaded shape is the signature of every exponential function: a long, quiet start that looks almost linear, then a wall. Tripling is the same wall, steeper — 3¹⁰ is already 59,049. Nothing in ordinary arithmetic prepares intuition for this, which is why the functions deserve their own tools and why this page limits itself to evaluating single powers exactly.

One base deserves naming because it owns the continuous version of the story: e ≈ 2.718282. e¹ is 2.718282, e² is 7.389056, and e^x is the function whose rate of growth at every point equals its own value — the reason it appears wherever growth is smooth rather than stepped. Where growth is stepped period by period and measured in money — an interest rate applied year after year — the calculation grows a principal column and a rate column and becomes the Compound Interest Calculator‘s territory; the pure function lives here and there, and the two pages deliberately do not duplicate each other.

The exponent laws, summarised

Eight rules — all of them the multiplication chain, re-read.

None of the laws of exponents is a new fact. Each one is the definition from the top of this page — an exponent counts multiplications — applied to a slightly larger expression and then noticed. The product rule is the clearest: 2³ × 2⁴ is (2 × 2 × 2) × (2 × 2 × 2 × 2), seven factors of 2 standing in a row, so of course the answer is 2⁷; the rule “add the exponents” is just counting the combined chain. The quotient rule subtracts because division cancels factors from the chain. The power-of-a-power rule multiplies because (2³)² is two chains of three. The summary below states all eight the compact way; treat the right-hand column as the receipt — every law was checked against the numbers it claims to govern before it was allowed into the table.

The eight laws of exponents, each with one computed check. The companion guide works every law slowly, with the standard wrong answers named.

Two disciplines keep the laws from being misapplied, and both are about scope. The product and quotient rules require the same base: 2³ × 2⁴ collapses into a single power, but 2³ × 3² does not collapse into anything — it is simply 8 × 9 = 72, computed head-on. And the power-of-a-product rule distributes over multiplication, not addition: (2 × 3)⁴ splits into 2⁴ × 3⁴, but (2 + 3)² does not split at all, as the mistakes section below shows with live numbers. The companion article to this page works each law once at full length — statement, chain of multiplications, check, and the wrong answer students most often write — for anyone who wants the rules to stick rather than merely to be available.

Common mistakes

Four ways powers go wrong on paper — each shown wrong, then right.

Multiplying the exponents instead of adding them

The product rule adds exponents; it never multiplies them. 2³ × 2⁴ is 2⁷ = 128, because the chains join end to end, seven factors in all. The tempting wrong move — multiply 3 × 4 and write 2¹² — gives 4,096, thirty-two times too big. If your answer to a same-base multiplication feels enormous, this is the error to check first.

Splitting a power across a sum

(a + b)² is not a² + b². With live numbers: (2 + 3)² = 5² = 25, while 2² + 3² = 4 + 9 = 13. The missing piece is the cross term — expanding properly, (a + b)² = a² + 2ab + b², and here 2 × 2 × 3 = 12 is exactly the gap between 13 and 25. Powers distribute over multiplication; over addition they do not, and the difference is the whole of early algebra.

Losing the sign on a negative base

Parentheses decide everything here. (−3)² means (−3) × (−3) = 9: an even count of negatives, a positive result. But −3², written without parentheses, means −(3 × 3) = −9, because the power binds to the 3 before the minus sign is applied. The calculator sidesteps the ambiguity by taking the base exactly as you type it — type −3 as the base and it squares −3 — but on paper the two writings are different questions, and exam markers know it.

Reading a tower of powers the wrong way

2^(3²) and (2³)² look like twins and are not. Towers are read from the top down: 3² = 9 first, so 2^(3²) = 2⁹ = 512. The parenthesised version groups from the bottom: (2³)² = 8² = 64. An eightfold difference from nothing but grouping — the cheapest marks in any test, lost or kept by one pair of brackets.

Related guides

Three deep-dives from this page’s guide: the laws worked slowly, the two extensions decoded, and the complete powers tables.

Related calculators

More tools for powers, roots and number sense.