Exponent Calculator

bⁿ = b × b × … × b (n factors) b⁻ⁿ = 1 ÷ (b × b × … × b)

b

The base, the number being multiplied by itself. It is a plain number with no unit, and it is allowed to be negative, decimal or zero — each of those has its own rule further down. A negative base is only valid with a whole-number exponent, because an even root of a negative number is not real

n

The exponent, the number of times the base appears in the product. It is a real number on this page, not just an integer: a fraction means a root (an exponent of 0.5 is the square root, of 0.3333 is close to the cube root), and a negative sign means the reciprocal of the positive power

bⁿ

The power itself, the primary reading. It is printed in scientific notation once it leaves the range from 0.001 up to a million, so 2 to the 64th shows as a mantissa and an exponent of ten rather than as a 20-digit integer. Nothing is capped here — the only ceiling is what a double-precision number can hold

b × b × … × b

The expanded form, the same expression written out as repeated multiplication. Products longer than six factors are abbreviated with an ellipsis; below that they are written in full, which is why 3 to the 4th shows four factors and 2 to the 10th does not show ten

b⁻ⁿ = 1 ÷ (b × b × … × b)

What a negative exponent does: it takes the reciprocal of the positive power, one divided by the product. 2 to the −3 is 1 ÷ 8, not −8. This is the single most common misreading of the notation, and it is why the expanded form is worth printing — the reciprocal is visible in it and nowhere else on the page

b⁰ = 1

Any non-zero base to the power of zero is 1, and the expanded form says so by printing a bare 1 rather than an empty product. The step-down rule forces it — each division by the base has to keep working as the exponent falls by one, and b ÷ b is 1 — so 7 to the 0 being 1 is not a convention the page chose. Zero to the zero is the one case that rule cannot settle, and it is refused

0.5 as an exponent

A fractional exponent is a root: raising a number to 0.5 is the square root, to one third is the cube root. This is what makes the page refuse a negative base with a fractional exponent — the cube root of a negative number is perfectly real, but computing it that way is not, and the root calculator is the page that knows the difference

Use it whenever a number is being multiplied by itself a fixed number of times and the answer is what you are after: compound growth written as a single factor rather than a chain of multiplications, a doubling repeated a set number of times, a scale factor applied again and again, a probability of the same event happening in a row, the size of a storage unit as a power of two. It is also the page for the two notations people mix up — the product written out and the power written short — because it prints both at once: the expanded form is the definition and the power is the value, and seeing them side by side is what makes the negative exponent readable. Use the root calculator when the exponent is the unknown and a root is what is being asked for, since the negative and fractional cases there have rules of their own. Use the exponential growth calculator when the base is fixed and derived from a rate, which is the shape of every interest and population question. Use the log calculator when the power is known and the exponent is what you want; that is the same relation read backwards.

1. 2 to the power of 10

   1. The base is 2 and the exponent is 10, so ten factors of 2 are multiplied together
   2. Pairwise: 2 × 2 = 4, and 4 × 4 = 16, and 16 × 16 = 256, which is eight factors
   3. Two factors remain: 256 × 2 × 2 = 1024
   4. The product has more than six factors, so the expanded form is abbreviated to 2 × 2 × … × 2

   The row the reference table ends on, and the one number in this batch worth knowing by heart: a kilobyte is 1024 bytes because of it. The abbreviation in the second reading is not a shortage of space — ten factors and an ellipsis carry exactly the same information as ten factors written out, and the count is already in the exponent you typed.

2. 2 to the power of −3

   1. The exponent is negative, so the answer is the reciprocal of the positive power
   2. The positive power first: 2 × 2 × 2 = 8
   3. The reciprocal of 8 is 1 ÷ 8 = 0.125
   4. The expanded form prints the reciprocal itself, so the division is visible: 1 ÷ (2 × 2 × 2)

   The case the expanded form earns its place on. A reader who takes the minus sign to mean a negative answer expects −8 and gets 0.125 instead; the reading 1 ÷ (2 × 2 × 2) is what makes the reciprocal obvious rather than something to be remembered. It is also the reading that says the answer had to come out smaller than 1, since dividing one by a number above one can do nothing else.

3. 1.05 to the power of 4

   1. Four factors of 1.05 are multiplied together, which is four years of 5 percent growth
   2. Two factors: 1.05 × 1.05 = 1.1025
   3. Four factors: 1.1025 × 1.1025 = 1.21550625
   4. The power is printed to six decimals, so it reads 1.215506

   The shape this page is used in most often outside a classroom: a decimal base just above 1, an exponent counting periods. Four factors give 1.2155, which is the same as saying the quantity grew by 21.55 percent over the four periods — the growth rate itself is what the exponential growth calculator derives the base from, and it reports that percentage directly instead of leaving you to subtract one.

4. 7 to the power of 0

   1. A zero exponent means the product is empty — no factors of 7 are multiplied at all
   2. An empty product is 1, which is the identity for multiplication: multiplying by it changes nothing
   3. The step-down rule agrees: 7¹ ÷ 7 = 1, and 7⁰ ÷ 7 = 7⁻¹, so 7⁰ has to be 1 for the pattern to hold
   4. The expanded form prints the bare 1 rather than an empty multiplication

   The entry most often expected to be 0 and most often guessed as such. Nothing is being multiplied, so nothing can come out of it — but 1 is the value that leaves every neighbouring power correct, which is why the definition is an empty product rather than an appeal to intuition. The reference table opens on 2⁰ = 1 for the same reason.

5. −2 to the power of 3

   1. The base is negative and the exponent is a whole number, which is the combination that is allowed
   2. −2 × −2 = 4, since multiplying two negatives gives a positive
   3. 4 × −2 = −8
   4. The count of negative factors is odd, so the sign of the answer is negative

   A negative base is not refused — only a negative base with a fractional exponent is, because that is where the answer stops being a real number. The sign of the result follows the parity of the exponent, which the expanded form makes countable: three factors of −2 pair up once and leave one negative factor over. Change the exponent to 4 and the answer is +16.

This page raises a number to a power and nothing else. It does not add powers, multiply them, or combine two of them — the laws of exponents are stated in the formula block and applied by hand, which is deliberate, since the page has no second base to apply them to. It does not take logarithms: given a power and a base, working out the exponent is the reverse question and belongs to the log calculator. A negative base is accepted only with a whole-number exponent, and the refusal is not an inability to compute — the cube root of −8 really is −2 — but a routing decision, because the arithmetic that would produce it is a root rather than a power and the root calculator is the page with that rule in it. The result is computed in double precision and is exact only where double precision is: whole numbers are stored exactly up to 2 to the 53rd, and past that a whole number comes back with neighbouring integers indistinguishable from it. The exponent is bounded at ±64 and the base at a million, which are display ranges rather than mathematical limits — the real ceiling is the point where the result stops being a representable number, and reaching it raises an error instead of printing an infinity. The expanded form is abbreviated above six factors, so it is a reading of the multiplication rather than a full transcription of it.

What does a negative exponent mean?

It means the reciprocal of the positive power, not a negative number. 2 with an exponent of −3 is 1 ÷ (2 × 2 × 2), which is 0.125 — the answer is smaller than 1, not below zero. The expanded form on this page prints the division rather than only the result, because the minus sign is the one piece of the notation that is routinely read backwards.

Why is anything to the power of zero equal to 1?

Because a zero exponent means an empty product, and an empty product is the identity for multiplication, which is 1. The step-down rule says the same thing: each time the exponent falls by one the value is divided by the base, and taking 7¹ = 7 down one step gives 7 ÷ 7 = 1. It is never 0 — nothing was multiplied, so nothing can come out of it.

What is 0 to the power of 0?

It has no defined value, and this page says so rather than printing one. Two rules collide on it: every non-zero base to the zero is 1, and zero to any positive exponent is 0. JavaScript returns 1 for this expression, but that is a decision made by the language for consistency across its own operations rather than a mathematical result, which is why the page refuses it instead of quietly agreeing.

Can I use a fractional exponent such as 0.5?

Yes, with a positive base — an exponent of 0.5 is the square root, so 2 to the 0.5 is 1.414214. A negative base with a fractional exponent is refused, because that answer is not a real number: the arithmetic behind a fractional power goes through a logarithm, and the logarithm of a negative number does not exist in the reals. Odd roots of negative numbers are real, and the root calculator is the page that knows the difference.

Why does the expanded form stop at six factors?

Because past that the ellipsis carries exactly as much information as the full product. 2 to the 64th written out is more than a hundred characters of the same repeated factor, and the count is already in the exponent you typed, so abbreviating to 2 × 2 × … × 2 loses nothing. Below six factors the multiplication is written in full, which is why 3 to the 4th shows all four of them.

When does the answer switch to scientific notation?

When it leaves the range from 0.001 up to a million. Inside that window the number is printed as an ordinary decimal, because a reader takes in 9,375 at a glance and would have to translate 9.375 × 10³ first. Outside it the mantissa and the exponent of ten are both shown, so 2 to the 64th reads as 1.844674 × 10¹⁹ rather than as a twenty-digit integer with no visible magnitude.

• Power — the definition of a power as repeated multiplication, together with the rules for negative and fractional exponents — Wolfram MathWorld (United States)
• Exponentiation — the operation itself, including the reciprocal rule for negative exponents and the treatment of the indeterminate case 0⁰ — Wolfram MathWorld (United States)
• Exponentiation — the arithmetic of powers across number systems, and the reason a negative base with a non-integer exponent has no real value — Wikipedia