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.
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. 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 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.
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.
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.