A power is a shorthand for repeated multiplication. The expression bn means multiplying the baseb by itself n times — so 24=2×2×2×2=16. The number b is the base, and n is the exponent (or power).
Powers compress long products into compact notation. Instead of writing 10×10×10×10×10×10, you write 106. The notation scales effortlessly: 10100 would take a page to write out as a product but fits in three characters as a power.
This tool builds a table of powers for any base from 2 to 10, letting you see the exact value, the multiplication that produces it, and how the values grow row by row. The default state — base 2, max power 10 — shows the classic doubling sequence that underlies binary numbers.
For full theory of exponents, exponent rules, and properties, see exponents and powers.
Setting Base and Max Power
Two inputs control the table: Base (the number being multiplied) and Max Power (the highest exponent shown). The base accepts integers from 2 to 10. Max Power accepts integers from 0 upward, with caps that depend on the base.
Type a value in either input. The table regenerates instantly, showing every row from b0 up to bmax. The default state — base 2, max power 10 — produces the doubling sequence 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024.
The Reset All button restores defaults. Invalid input (out of range, non-numeric, or empty) shows an inline error and outlines the input in red until corrected. The error message tells you exactly what went wrong: minimum base is 2, maximum base is 10, minimum power is 0, and the maximum power depends on which base is currently selected.
If you change the base from a small value to a larger one and your current max power exceeds the new cap, the tool automatically adjusts max power down to the new limit so the table stays valid.
Reading the Three Columns
Each row of the table represents one power and breaks it into three pieces.
The Power column shows the notation bn. The base appears full size with the exponent as a superscript — exactly how you would write it on paper. The exponent counts down the rows from b0 at the top to bmax at the bottom.
The Expression column spells out the multiplication: b×b×b×…. For exponents up to 10, every factor is shown explicitly. For exponents above 10, the expression abbreviates to b×b×⋯×b(ntimes) so the row stays readable. The first row, with exponent 0, simply shows the value 1, since there is no multiplication to spell out.
The Value column gives the exact numerical result. Values use thousand separators for readability, and large numbers are computed with arbitrary-precision integer arithmetic — there are no rounding errors even at b16.
Below the table, the pattern note reminds you of the row-to-row relationship: each row equals the previous row multiplied by the base.
Exploring Different Bases
Each base produces a distinctively shaped sequence. Switching bases is the fastest way to build intuition for how exponential growth depends on the base, not just the exponent.
Base 2 doubles each row: 1, 2, 4, 8, 16, .... Powers of 2 underlie binary representation, file sizes (kilobyte, megabyte, gigabyte), and combinatorics — the number of subsets of an n-element set is exactly 2n. Small bases also unlock a longer table — see pushing to maximum powers.
Base 10 produces place value: 1, 10, 100, 1{,}000, .... Each row adds a zero. The decimal number system is built directly on this progression, which is why scientific notation uses powers of 10.
Base 3 through base 9 fill in the gaps. Base 5 is the first base the depth cap applies to, and base 7 has the most interesting last-digit cycle. Powers of 3 grow faster than powers of 2 but slower than 10. Powers of 9 reach into the billions by exponent 10: 910=3,486,784,401. Switch the base back and forth to compare growth rates side by side at the same maximum exponent.
The Default Table: Powers of Two
The tool's opening state: base 2, max power 10 — eleven rows running 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1,024. The doubling sequence, exact and fully spelled out.
Base 2, max power 10, frozen
The opening doubling sequence, 1 to 1,024 in eleven rows — every factor of every expression spelled out.
Base 2 is the default for a reason: it is the smallest possible base, so its table grows the slowest and stays the most readable — and it happens to be the most consequential sequence in modern life. The rows are the file sizes, the memory ranges, and the subset counts: 210=1,024 is the "kilo" of kilobyte, and a set of ten elements has exactly 1,024 subsets.
The Expression column earns its keep here more than anywhere: watching 2×2×2×2 grow one factor per row makes the definition of exponentiation visible rather than notational. Every claim in the theory sections below — the zero-power rule, the ×base row law — can be checked against these eleven rows by eye. When the doubling stops feeling surprising, push deeper: the sixteen-row table is one input away.
Powers of Ten and Place Value
Base 10, max power 10: the table where the Value column writes itself — 1, 10, 100, 1,000, and so on to 1010=10,000,000,000, one zero added per row.
Base 10, max power 10, frozen
Place value as a table: one zero per row, ten billion by the bottom — the ladder scientific notation climbs.
This is the table you already know by heart, which is exactly what makes it worth freezing: the decimal number system is this table. Writing "4,507" means 4⋅103+5⋅102+0⋅101+7⋅100 — place value is powers of ten wearing digit clothing, and the thousand separators in the Value column mark off the rows of this very table.
Ten is also the largest base the tool accepts, which makes this the fastest-growing table on offer: ten billion by row ten. Scientific notation exists precisely to ride this ladder — 3×108 names a rung rather than spelling out zeros. Compare the default base-2 table at the same ten rows: 1,024 versus 10,000,000,000, the whole answer to "does the base matter?"
Base Five at the Cap Boundary
Base 5, max power 10: the first base the depth cap applies to — the Max Power input tops out at 10 here, where base 4 would allow 16. The table climbs to 510=9,765,625.
Base 5, max power 10, frozen
The first capped base at its deepest: 9,765,625 at row ten, and a Value column ending in 5 all the way down.
The boundary itself is the state's first lesson: the cap rule (bases 2–4 go to power 16, bases 5–10 stop at 10) is about display honesty, not mathematical limits. 516 has twelve digits; the tool prefers a shorter table it can render cleanly to a longer one it cannot. Try entering 16 with base 5 loaded and the input politely refuses — the frozen frame shows the deepest base-5 view that exists.
The second lesson is in the Value column's right edge: every row past 50 ends in the digit 5 — the simplest of all the last-digit patterns, a cycle of length one. Multiplying a number ending in 5 by 5 always yields a number ending in 5, so the pattern locks in forever. The richer cycles this previews are the subject of spotting last-digit patterns.
The Zero Power Rule
The first row of every table — regardless of base — shows b0=1. This is not a quirk; it is a rule that keeps exponent arithmetic consistent.
The exponent law bm÷bn=bm−n requires b0=1 for any nonzero base. Setting m=n gives b0 on one side, while the same expression also equals any nonzero bn/bn=1. Both must agree, forcing b0=1.
The Expression column displays this row simply as the value 1 — there is no multiplication to spell out, since the rule says zero copies of the base produce the multiplicative identity. Every base from 2 to 10 obeys this rule, which is why every powers table opens the same way.
Pushing to Maximum Powers
For small bases (2 through 4), the table allows exponents up to 16. For larger bases (5 through 10), the maximum exponent is capped at 10. The cap exists because high powers of large bases produce numbers with too many digits to display readably in a table cell.
Hover over the ? icon next to the Max Power input to see the rule spelled out. The tooltip explains the cap and reminds you that bases 2 through 4 give you the longer range.
At the maximum, base 2 reaches 216=65,536 — the value 64K, familiar from old computers and 16-bit integer ranges. Base 3 reaches 316=43,046,721. Base 4 reaches 416=4,294,967,296 — the same as 232, the size of the 32-bit unsigned integer space. Even with the cap, bases 5 through 10 comfortably handle numbers in the billions and tens of billions.
Base 2, max power 16, frozen
Seventeen rows to 65,536: the abbreviated expression past exponent 10, and half a percent of the value reached by the halfway row.
The frozen table above shows the deepest view the tool offers: base 2 pushed to all seventeen rows, from 20=1 to 216=65,536. Two rendering rules become visible only here. The Expression column switches to its abbreviated form past exponent 10 — 2×2×⋯×2(16times) — because seventeen explicit factors would overflow the cell. And the values stay exact to the last digit, courtesy of the arbitrary-precision arithmetic underneath.
Sixteen doublings is also a good place to feel the exponential law: halfway down the table, at 28, the value is only 256 — under half a percent of the final value. Exponential sequences spend most of their journey looking small; the last few rows do most of the growing. Compare the default ten-row table to see how different the same base looks when cut six rows shorter.
Spotting Last-Digit Patterns
Powers cycle through predictable last digits. The Value column makes the cycles visible at a glance — scan the rightmost digit of each row and the cycle reveals itself.
Base 5: every power except 50 ends in 5. The pattern is constant: 1, 5, 25, 125, 625, 3{,}125, .... This makes mental arithmetic with powers of 5 unusually easy.
Base 7: last digits cycle through 7, 9, 3, 1 with period 4. So 71 ends in 7, 72=49 ends in 9, 73=343 ends in 3, 74=2,401 ends in 1, then 75 ends in 7 again. The cycle repeats forever.
Base 4 alternates between last digits 4 and 6. Base 9 alternates between 9 and 1. Base 6 stays at 6 once you pass 60.
Every base from 2 to 9 has a finite last-digit cycle. Only base 10 breaks the pattern — every power 10n for n≥1 ends in 0. Switching bases and watching the rightmost column is one of the fastest ways to feel the structure of modular arithmetic.
Base 7, max power 10, frozen
The last-digit laboratory: 7, 9, 3, 1 cycling down the rightmost digits, twice around and a half in ten rows.
The frozen frame above fixes base 7 — the base with the richest cycle — so the pattern can be traced down a real Value column: 1, 7, 49, 343, 2,401, 16,807, ... with last digits running 7, 9, 3, 1 and returning to 7 at 75. Period four, twice around and a half in ten rows.
The cycle exists because the last digit of a product depends only on the last digits of its factors — multiplication survives reduction mod 10. Each row's final digit is therefore determined by the previous row's final digit alone, and since there are only ten possible digits, the sequence must eventually revisit one and loop forever. That little argument — finitely many states force a cycle — is the pigeonhole principle doing number theory, and the table is its laboratory.
Why Powers Grow So Fast
Exponential growth is genuinely different from multiplication or repeated addition. Each row of the table multiplies — not adds — by the base. Compare 210=1,024 to 910=3,486,784,401: the same exponent, but the larger base produces a number more than three million times bigger.
The pattern note at the bottom of the table summarizes it: each row is ×base the row above. After 10 rows, base 2 has multiplied by 210=1,024 — roughly a thousandfold. Base 10 has multiplied by 1010 — ten billion. Base 9 sits between them, but much closer to base 10 than to base 2.
This is why exponential growth shows up in population dynamics, compound interest, viral spread, and computer science complexity. A linear process gains a fixed amount per step; an exponential process multiplies by a fixed factor per step, and the gap between the two opens up rapidly.
For the formal definition and behavior of exponential functions, see exponential function.
Related Concepts and Tools
Exponents and Powers — Theory of exponents, including negative powers, fractional exponents, and the laws of exponents.
Exponential Function — How bx behaves as a continuous function of x, including the special role of base e.
Logarithms — The inverse operation of exponentiation: if bn=v, then logbv=n. Use logarithms to find the exponent given the base and the value.
Algebraic Identities Visualizers — Geometric proofs of squared identities like (a+b)2 and (a−b)2, showing where powers of 2 appear in algebra.
Algebra Visual Tools — Index of all interactive tools for algebra topics including identity proofs and the powers table itself.