Exponents in simple words
An exponent is a small number written up and to the right of another number, like the 3 in 2³. It tells you how many copies of the bottom number (the base) to multiply together.
Example: 2³ means 2 × 2 × 2. First 2 × 2 = 4, then 4 × 2 = 8. So 2³ = 8. In the same way, 5² = 5 × 5 = 25, which is why a power of 2 is called “squared”.
A power of 0 always gives 1 (for any base except 0), so 7⁰ = 1. A negative exponent means 1 divided by the power: 2⁻³ = 1 ÷ 8 = 0.125.
On a computer we often type the exponent after a ^ sign, so 2^3 means 2³.
Powers, reciprocals and roots
A positive integer exponent means repeated multiplication: 2¹⁰ = 1,024. A negative exponent gives a reciprocal, so 2⁻³ = 1 ÷ 8 = 0.125. For positive bases, a fractional exponent describes a root: 9^0.5 = 3.
To solve b^e = r for the exponent, use e = ln(r) ÷ ln(b), requiring b > 0, b ≠ 1 and r > 0. To find a principal base for a non-zero exponent, use r^(1/e), subject to the real-number domain.
Example: recover the base and exponent
For b³ = 125, the base is 125^(1/3) = 5. For 2^e = 1,024, e = ln(1,024) ÷ ln(2) = 10. An even integer exponent may have two real bases: b² = 25 has b = +5 and b = −5. The calculator shows both signs and uses the positive base for its reference table.
Real-number and precision limits
Negative bases with integer exponents work, including odd roots of negative results in base-solving mode. Decimal exponents on negative bases are unsupported because a rounded decimal does not identify an exact rational root. Type the exponent as a fraction instead, such as 1/3: (−8)^(1/3) = −2.
Zero to a positive exponent is zero. This tool treats 0^0 and zero to a negative exponent as undefined, and cannot recover a unique base from exponent zero. The logarithm mode does not solve negative-base equations. Overflow is rejected; very small powers may underflow to zero. Results use floating-point arithmetic rather than symbolic exact algebra.
Fractional exponents
Type an exponent as a fraction m/n to use the rule b^(m/n) = (ⁿ√b)^m. For 8^(2/3), ∛8 = 2 and 2² = 4. A negative exponent adds a reciprocal: 27^(−2/3) = 1 ÷ (∛27)² = 1/9.
The fraction is reduced to lowest terms first. A negative base then has a real value only when the denominator is odd: (−8)^(1/3) = −2, but (−8)^(1/2) is not real. When the root is irrational, the exact radical is also shown, such as 2^(3/2) = √8 = 2√2.
The first 100 perfect cubes
A perfect cube is an integer raised to the power 3: n³ = n × n × n. Reading the table backwards gives exact cube roots, so ∛2,197 = 13 because 13³ = 2,197. Cubes keep the sign of n, so (−4)³ = −64 and ∛(−64) = −4.
The last digit of n³ decides the last digit of n uniquely (0↔0, 1↔1, 2↔8, 3↔7, 4↔4, 5↔5, 6↔6, 7↔3, 8↔2, 9↔9), which is a quick check on a cube root. Enter any base with exponent 3 above to cube a number beyond 100.
| n | n³ | Written out |
|---|---|---|
| 1 | 1 | 1 × 1 × 1 |
| 2 | 8 | 2 × 2 × 2 |
| 3 | 27 | 3 × 3 × 3 |
| 4 | 64 | 4 × 4 × 4 |
| 5 | 125 | 5 × 5 × 5 |
| 6 | 216 | 6 × 6 × 6 |
| 7 | 343 | 7 × 7 × 7 |
| 8 | 512 | 8 × 8 × 8 |
| 9 | 729 | 9 × 9 × 9 |
| 10 | 1,000 | 10 × 10 × 10 |
| 11 | 1,331 | 11 × 11 × 11 |
| 12 | 1,728 | 12 × 12 × 12 |
| 13 | 2,197 | 13 × 13 × 13 |
| 14 | 2,744 | 14 × 14 × 14 |
| 15 | 3,375 | 15 × 15 × 15 |
| 16 | 4,096 | 16 × 16 × 16 |
| 17 | 4,913 | 17 × 17 × 17 |
| 18 | 5,832 | 18 × 18 × 18 |
| 19 | 6,859 | 19 × 19 × 19 |
| 20 | 8,000 | 20 × 20 × 20 |
| 21 | 9,261 | 21 × 21 × 21 |
| 22 | 10,648 | 22 × 22 × 22 |
| 23 | 12,167 | 23 × 23 × 23 |
| 24 | 13,824 | 24 × 24 × 24 |
| 25 | 15,625 | 25 × 25 × 25 |
| 26 | 17,576 | 26 × 26 × 26 |
| 27 | 19,683 | 27 × 27 × 27 |
| 28 | 21,952 | 28 × 28 × 28 |
| 29 | 24,389 | 29 × 29 × 29 |
| 30 | 27,000 | 30 × 30 × 30 |
| 31 | 29,791 | 31 × 31 × 31 |
| 32 | 32,768 | 32 × 32 × 32 |
| 33 | 35,937 | 33 × 33 × 33 |
| 34 | 39,304 | 34 × 34 × 34 |
| 35 | 42,875 | 35 × 35 × 35 |
| 36 | 46,656 | 36 × 36 × 36 |
| 37 | 50,653 | 37 × 37 × 37 |
| 38 | 54,872 | 38 × 38 × 38 |
| 39 | 59,319 | 39 × 39 × 39 |
| 40 | 64,000 | 40 × 40 × 40 |
| 41 | 68,921 | 41 × 41 × 41 |
| 42 | 74,088 | 42 × 42 × 42 |
| 43 | 79,507 | 43 × 43 × 43 |
| 44 | 85,184 | 44 × 44 × 44 |
| 45 | 91,125 | 45 × 45 × 45 |
| 46 | 97,336 | 46 × 46 × 46 |
| 47 | 103,823 | 47 × 47 × 47 |
| 48 | 110,592 | 48 × 48 × 48 |
| 49 | 117,649 | 49 × 49 × 49 |
| 50 | 125,000 | 50 × 50 × 50 |
| 51 | 132,651 | 51 × 51 × 51 |
| 52 | 140,608 | 52 × 52 × 52 |
| 53 | 148,877 | 53 × 53 × 53 |
| 54 | 157,464 | 54 × 54 × 54 |
| 55 | 166,375 | 55 × 55 × 55 |
| 56 | 175,616 | 56 × 56 × 56 |
| 57 | 185,193 | 57 × 57 × 57 |
| 58 | 195,112 | 58 × 58 × 58 |
| 59 | 205,379 | 59 × 59 × 59 |
| 60 | 216,000 | 60 × 60 × 60 |
| 61 | 226,981 | 61 × 61 × 61 |
| 62 | 238,328 | 62 × 62 × 62 |
| 63 | 250,047 | 63 × 63 × 63 |
| 64 | 262,144 | 64 × 64 × 64 |
| 65 | 274,625 | 65 × 65 × 65 |
| 66 | 287,496 | 66 × 66 × 66 |
| 67 | 300,763 | 67 × 67 × 67 |
| 68 | 314,432 | 68 × 68 × 68 |
| 69 | 328,509 | 69 × 69 × 69 |
| 70 | 343,000 | 70 × 70 × 70 |
| 71 | 357,911 | 71 × 71 × 71 |
| 72 | 373,248 | 72 × 72 × 72 |
| 73 | 389,017 | 73 × 73 × 73 |
| 74 | 405,224 | 74 × 74 × 74 |
| 75 | 421,875 | 75 × 75 × 75 |
| 76 | 438,976 | 76 × 76 × 76 |
| 77 | 456,533 | 77 × 77 × 77 |
| 78 | 474,552 | 78 × 78 × 78 |
| 79 | 493,039 | 79 × 79 × 79 |
| 80 | 512,000 | 80 × 80 × 80 |
| 81 | 531,441 | 81 × 81 × 81 |
| 82 | 551,368 | 82 × 82 × 82 |
| 83 | 571,787 | 83 × 83 × 83 |
| 84 | 592,704 | 84 × 84 × 84 |
| 85 | 614,125 | 85 × 85 × 85 |
| 86 | 636,056 | 86 × 86 × 86 |
| 87 | 658,503 | 87 × 87 × 87 |
| 88 | 681,472 | 88 × 88 × 88 |
| 89 | 704,969 | 89 × 89 × 89 |
| 90 | 729,000 | 90 × 90 × 90 |
| 91 | 753,571 | 91 × 91 × 91 |
| 92 | 778,688 | 92 × 92 × 92 |
| 93 | 804,357 | 93 × 93 × 93 |
| 94 | 830,584 | 94 × 94 × 94 |
| 95 | 857,375 | 95 × 95 × 95 |
| 96 | 884,736 | 96 × 96 × 96 |
| 97 | 912,673 | 97 × 97 × 97 |
| 98 | 941,192 | 98 × 98 × 98 |
| 99 | 970,299 | 99 × 99 × 99 |
| 100 | 1,000,000 | 100 × 100 × 100 |