Any base to any power: the exact answer, scientific notation when it is huge, and the rule that got there
Enter a base and an exponent to raise one to the other. Whole-number powers are worked out exactly, even when the answer runs to thousands of digits, so 9 to the power 81 is shown in full rather than rounded. Negative exponents give an exact fraction, fractional exponents are treated as roots, and fractions such as 5/6 can be used as the base. Each answer comes with the rule that produced it, which is the part worth remembering for the next question.
An exponent counts how many copies of the base are multiplied together. 2 to the power 13 is thirteen 2s multiplied, which is 8,192; 25 squared is 25 × 25, which is 625. Squared means to the power 2 and cubed means to the power 3, because those are the area of a square and the volume of a cube with that side length.
Powers of 10 are the easiest to read: the exponent is the number of zeros. 10 to the power 18 is a 1 followed by eighteen zeros, a quintillion, and that is exactly why scientific notation writes very large numbers as a value between 1 and 10 multiplied by a power of 10.
A negative exponent means one over the positive power: 2 to the power −3 is 1 ÷ 2³, which is 1/8. It never makes the answer negative. Any non-zero number to the power 0 is 1, which follows from the division rule: 2⁵ ÷ 2⁵ is both 1 and 2 to the power 5 − 5.
The bottom of a fractional exponent is a root and the top is a power. 16 to the power 1/2 is the square root of 16, which is 4; 27 to the power 2/3 is the cube root of 27, squared, which is 9. When the root is not a whole number the answer is irrational, so it is shown as a decimal.
A negative number has no real square root, so a negative base with an even root, such as −4 to the power 1/2, has no real answer. An odd root is fine: the cube root of −8 is −2.
Multiplying powers of the same base adds the exponents: 2³ × 2⁴ = 2⁷. Dividing subtracts them: 2⁷ ÷ 2³ = 2⁴. A power of a power multiplies them: (2³)² = 2⁶. A power of a fraction applies to top and bottom: (5/6)² = 25/36, which is why (5/6) to the power 30 shrinks towards zero — the bottom grows faster than the top.
8,192. It is 2 multiplied by itself 13 times; 2¹⁰ is 1,024 and 2³ is 8, and 1,024 × 8 = 8,192.
625, because 25 × 25 = 625. A quick way: numbers ending in 5 square to (tens × next number) followed by 25, so 2 × 3 = 6, giving 625.
1,000,000,000,000,000,000 — a 1 followed by eighteen zeros, called a quintillion in English. In scientific notation it is 1 × 10¹⁸.
One over the positive power. 2⁻³ = 1/2³ = 1/8, and 10⁻² = 1/100 = 0.01. The answer gets smaller, not negative.
Raise the top and the bottom separately. (5/6)² = 25/36. With large exponents the answer shrinks quickly because the bottom grows faster: (5/6)³⁰ is about 0.0042.
Calculators and algebra treat it as 1, because it keeps the rules for powers and polynomials consistent. Some parts of calculus leave it undefined, so a textbook may say either.