Exponent Calculator

What a power means

An exponent counts how many copies of a number you multiply together. In 2^10 the base is 2 and the exponent is 10, so the power is ten 2s multiplied: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 1,024. The small raised number is a shorthand, not a separate value to multiply.

Two ways of reading the same expression help. One is repeated multiplication. The other is growth: each factor of 2 doubles what came before, so 2^10 is ten doublings. A bank account that grows 5 percent a year follows 1.05^n, where n counts the years.

Powers show up wherever a quantity multiplies by a fixed factor: compound interest, population, half-lives, decibels, earthquake magnitudes and the area of a square. The last one is where the language comes from. 5^2 is read as five squared because it is the area of a square with sides of 5, and 5^3 is five cubed because it is the volume of a cube with edges of 5.

The calculator takes any base, including decimals and negative numbers, and any exponent, including fractions and negatives. The second panel answers a different question: how many periods does a growth rate need to reach a target.

The exponent rules that cover most homework

Six rules handle nearly every problem that mixes powers. They all come from counting factors, so you can rebuild any of them on paper when you forget one.

The rules only combine powers that share a base. 2^3 x 3^4 does not simplify, because 2 and 3 are different factors. That single point causes more lost marks than any other, since the rule is about counting identical factors, not about multiplying numbers that happen to sit next to each other.

The zero rule looks odd until you divide. 2^3 / 2^3 is 8 / 8 = 1, and subtracting the exponents gives 2^0, so 2^0 has to be 1 for the rule to stay consistent. The same argument makes 5^0 and 100^0 equal to 1. Zero itself is the one exception, covered below. Wolfram MathWorld's list of exponent laws states each rule in the same order.

Worked example: 2^10 and a 5% growth rate

The calculator shows 2^10 as 1,024. Doubling five times gives 32, and doubling five more times gives 32 x 32 = 1,024. Splitting a power in half like that is faster than ten separate multiplications, and it is the same trick that makes the rule for multiplying powers useful.

Growth works the same way with a base slightly above 1. At 5 percent a year, money multiplies by 1.05 per year, so the balance after n years is the starting amount times 1.05^n. Starting with $1,000:

Notice the shape. The first ten years add $628.89, the second ten add $1,024.41, and the third ten add $1,668.64. The rate never changed. Each decade multiplies a larger balance, which is why the later years look so much stronger on a chart.

Two cautions. First, a 5 percent return is not a promise, and a single bad year changes the base for every year after it. Second, the order of returns matters once you add or remove money, so a compound growth formula describes a smooth rate that real accounts rarely deliver.

Negative and zero exponents

A negative exponent flips the number into a fraction. 2^-3 means 1 / 2^3, which is 1 / 8 = 0.125. The minus sign does not make the answer negative. It moves the power to the other side of the fraction bar, so 2^-3 is positive and (-2)^-3 is negative only because the base is negative.

Zero is the border between the two. 2^0 = 1 sits between 2^1 = 2 and 2^-1 = 0.5, and each step down divides by 2, so the sequence runs 2, 1, 0.5, 0.25, 0.125. Put another way, 10^-2 = 0.01, which is why scientific notation can write small numbers with a negative exponent.

Zero to the power zero is the one case with no agreed answer. Some fields define it as 1 for convenience, and others leave it undefined because the division argument breaks when the base is zero. This calculator reports it as undefined rather than picking a side, which matches what most textbooks expect from a student.

Zero to a positive power is simple: 0^5 = 0. Zero to a negative power has no value at all, since it would require dividing by zero.

Fractional exponents are roots

An exponent of 0.5 is a square root, an exponent of 1/3 is a cube root, and the denominator of the fraction names the root. So 9^0.5 = 3 and 27^(1/3) = 3. The top of the fraction is a normal power applied after the root, which is why 16^0.75 = 8: a quarter power is a fourth root, the fourth root of 16 is 2, and 2 cubed is 8.

You can take the root first or the power first and get the same answer. 16^0.75 can be read as the fourth root of 16 cubed, which is the fourth root of 4,096, and that is also 8. Roots first is easier by hand because the numbers stay small.

Negative bases break this. (-8)^0.5 asks for the square root of a negative number, which is not a real number, and the calculator says so instead of printing a result. A cube root of a negative number does exist, since -2 multiplied by itself three times gives -8. Browsers compute fractional powers with logarithms, so a negative base with a fractional exponent returns an error even when a real answer exists.

Powers of ten and scientific notation

Ten is the base that matters most in daily life, because decimal notation is built on it. 10^3 = 1,000 while 10^6 = 1,000,000, and 10^-3 = 0.001. Scientific notation pulls out one of those powers and leaves a number between 1 and 10 in front, so 6.02 x 10^23 is Avogadro's number and 3 x 10^-9 is three billionths.

The prefixes on units follow the same ladder. NIST publishes the SI prefix table, where kilo means 10^3, mega means 10^6 and giga means 10^9. The calculator converts every answer into that form, which is handy once a result runs past six digits.

Binary and decimal powers drift apart quickly. 2^10 = 1,024 is close to 10^3 = 1,000, which is why the two get confused in storage sizes. At 2^20 = 1,048,576 against 10^6 = 1,000,000 the gap is 4.86 percent, and it keeps growing with each step. That small drift is the source of the missing gigabytes on a new hard drive.

Doubling is a useful mental check on any power of two. Ten doublings are about three decimal digits, so 2^30 is close to 10^9, and 2^40 is close to 10^12.

Finding the exponent with logarithms

Sometimes the base and the answer are known and the exponent is the unknown. If a fund grows 5 percent a year, how long until it doubles? The equation is 1.05^n = 2, and a logarithm unwraps the exponent: n = ln(2) / ln(1.05) = 14.2067 years. Check it by raising 1.05 to that power, which returns 2.

The rule of 72 gives the same answer without a calculator. Divide 72 by the percentage rate: 72 / 5 = 14.4 years, close to the exact 14.21. The rule works because ln(2) is about 0.693 and ln(1 + r) is close to r for small r, so the exact answer is roughly 69.3 / r, which people rounded to 72 because it divides evenly by more numbers.

The same step answers other questions. The calculator's second panel does this for any growth factor and any target, and prints the check so you can see the round trip.

Logarithms only work this way when both sides are positive. A negative growth factor or a target of zero has no solution here, which is why the panel asks for a factor above zero.

Where calculator answers drift

Browsers store numbers in a fixed 64-bit format, and that format runs out of exact whole numbers at 2^53, which is 9,007,199,254,740,992. Every integer up to that value is exact. Above it, the gap between representable numbers grows to 2, and results start to lose their last digits.

The ceiling comes later. The largest finite value is about 1.8 x 10^308, so 10^400 overflows and the calculator reports that the number is too large instead of printing a wrong one. Between those two limits, results are correct to about 15 or 16 significant digits, which is plenty for money and measurements and not enough for exact work with huge integers.

Three cases deserve a second look before you trust the screen. A negative base with a fractional exponent is not a real number, so an error there is correct. Zero to the power zero is undefined. And an answer that ends in a string of zeros or nines may be a rounding artefact rather than an exact value, which is why the page shows a trimmed result instead of 17 digits.

For homework, the exponent rules in the table above are the part worth memorizing. The calculator is a check on the arithmetic, not a replacement for knowing which rule applies.