Exponent Calculator
Calculate bⁿ for whole, negative and decimal exponents, with scientific notation and a breakdown of what the power means, such as 10 factors of 2.
What is an exponent calculator?
An exponent tells you how many times a number, the base, is used as a factor. In 2¹⁰, the base is 2 and the exponent is 10, so 2¹⁰ is 10 factors of 2 multiplied together: 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1,024. Exponents are also called powers, and 2¹⁰ is read "2 to the 10th power".
This exponent calculator raises a Base (b) to an Exponent (n) and returns three things:
- bⁿ: the value, to 8 decimal places.
- Scientific notation: very large or very small results written as a number times a power of 10, such as 1.000000 × 10^9.
- What it means: a plain-language reading. For 2¹⁰ it says
10 factors of 2: 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024; for a negative exponent it shows the reciprocal, and for a power of 0.5 it shows the square root.
The exponent can be a whole number, a negative number or a decimal such as 0.5 or 2.5. The base can be negative when the exponent is a whole number, or a decimal that is exactly a fraction with an odd denominator, such as 0.2 = 1/5. Cases with no real answer, such as 0 to a negative power or an even root of a negative number, are labeled instead of producing an error, and 0⁰ is reported as undefined (see the questions below).
In the Common Core standards, students apply the properties of integer exponents and work with scientific notation in grade 8 (8.EE.A.1, 8.EE.A.3 and 8.EE.A.4), then extend exponents to fractions in high school (N-RN.A.1).
Formula
For a whole-number exponent n ≥ 1, the power is repeated multiplication with n factors:
bⁿ = b × b × … × b (n factors of b)
That is n factors, but only n − 1 multiplication signs: 2³ = 2 × 2 × 2. This is why the calculator says "10 factors of 2" for 2¹⁰ rather than "2 multiplied by itself 10 times", a phrase that could be read as 11 factors. The other exponents follow from keeping the rules consistent:
| Rule | Formula | Example |
|---|---|---|
| Zero exponent | b⁰ = 1 (b ≠ 0) | 5⁰ = 1 |
| Negative exponent | b⁻ⁿ = 1 ÷ bⁿ | 2⁻³ = 1 ÷ 8 = 0.125 |
| Product of powers | bᵐ × bⁿ = bᵐ⁺ⁿ | 3² × 3⁻⁵ = 3⁻³ = 1/27 |
| Quotient of powers | bᵐ ÷ bⁿ = bᵐ⁻ⁿ | 2⁷ ÷ 2⁴ = 2³ = 8 |
| Power of a power | (bᵐ)ⁿ = bᵐⁿ | (2³)² = 2⁶ = 64 |
| Fractional exponent | b1/n = ⁿ√b, bm/n = (ⁿ√b)ᵐ | 161/2 = 4, 95/2 = 3⁵ = 243 |
The zero rule comes from the quotient rule: bᵐ ÷ bᵐ = 1, and also bᵐ⁻ᵐ = b⁰. The fractional rule is how the Common Core standard N-RN.A.1 motivates rational exponents: 51/3 is the cube root of 5, because (51/3)³ must equal 5. With a negative base, the calculator writes a decimal exponent exactly as a fraction in lowest terms. An odd denominator is an odd root, which is real: (−32)0.2 = (−32)1/5 = −2. An even denominator is an even root of a negative number, which is not real: (−8)0.5 = √(−8).
Scientific notation
When the size of the result, ignoring any minus sign, is 1,000,000 or more, or less than 0.0001 but not 0, the Scientific notation line writes it as a number from 1 to 10 (with its sign) times a power of 10, with 6 decimal places and "^" for the power. For example, 10⁹ is shown as 1.000000 × 10^9 and 10⁻⁵ as 1.000000 × 10^-5. Other results are shown as ordinary numbers.
Worked example
Example 1: a whole-number exponent
Enter base 2 and exponent 10. bⁿ is 1,024 and What it means reads 10 factors of 2: 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024. Doubling ten times gives the same answer: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1,024. For exponents above 10 the line only gives the count, for example 11 factors of 2 for 2¹¹ = 2,048.
Example 2: a negative exponent
Enter base 2 and exponent −3. The result is 0.125, and What it means reads 2^-3 = 1 ÷ 2^3 = 1 ÷ 8 = 0.125. A negative exponent does not make the answer negative; it means "1 divided by". The same rule explains the grade 8 example 3² × 3⁻⁵ = 3⁻³: base 3 and exponent −3 give 3^-3 = 1 ÷ 3^3 = 1 ÷ 27 = 0.03703704, which is 1/27.
Example 3: a negative base
Enter base −3 and exponent 4. The calculator shows 4 factors of -3: (-3) × (-3) × (-3) × (-3) = 81. An even number of negative factors gives a positive result. With base −2 and exponent 3, the result is −8, because there are three negative factors.
Watch the notation on paper: (−3)⁴ = 81, but −3⁴ means −(3⁴) = −81, because the exponent applies before the minus sign. The calculator always treats the number in the Base box as the whole base, as if it were in parentheses.
Example 4: a fractional exponent
Enter base 9 and exponent 2.5. Since 2.5 = 5/2, 92.5 = (√9)⁵ = 3⁵ = 243. The calculator shows 243 and notes "(a half power is a square root)". Similarly, base 32 with exponent 0.6 gives 8: 0.6 = 3/5, the fifth root of 32 is 2, and 2³ = 8.
Example 5: a very large result
Base 7 with exponent 20 gives 7.979227 × 10^16 in Scientific notation. The full value in bⁿ is shown as 79,792,266,297,612,000.00000000, but the exact answer is 79,792,266,297,612,001. The calculator keeps about 16 significant digits, so above 2⁵³ = 9,007,199,254,740,992 even a whole-number power is not guaranteed exact, and its last digits can be rounded.
How to use this calculator
- Enter the Base (b), the number being multiplied. It can be negative or a decimal.
- Enter the Exponent (n). Use a decimal for a fractional power, for example 0.5 for a square root or 2.5 for 5/2.
- Read the value in bⁿ.
- For a very large or very small result, use Scientific notation, which keeps 7 significant digits.
- Read What it means to see the result as repeated factors, a reciprocal or a root.
Which output to use
- bⁿ is best for everyday sizes. It always shows 8 decimal places, so a tiny result such as 2⁻²⁰ appears as 0.00000095.
- Scientific notation keeps the detail in that case: 2⁻²⁰ is
9.536743 × 10^-7.
Common mistakes
- Multiplying base and exponent. 2¹⁰ is 1,024, not 2 × 10 = 20.
- Thinking a negative exponent gives a negative answer. 2⁻³ is 0.125.
- Dropping parentheses. (−3)⁴ = 81, while −3⁴ = −81.
- Rounding a fractional exponent too much. 1/3 cannot be typed exactly as a decimal. Base 27 with exponent 0.33 gives 2.96722201 and 0.3333 gives 2.99967043, not 3. Type more 3s, or use the square root calculator, which takes the root index as a whole number.
More help: the exponent rules cheat sheet, how to show full decimal numbers without scientific notation and the logarithm calculator, which answers the reverse question: what exponent gives a certain result.
Sources
- Common Core State Standards Initiative — Grade 8: Expressions & Equations (8.EE.A.1, 8.EE.A.3, 8.EE.A.4)
- Common Core State Standards Initiative — High School: Number and Quantity, The Real Number System (N-RN.A.1)
- NIST Digital Library of Mathematical Functions — §4.2 Definitions (powers, eqs. 4.2.26 and 4.2.27)
Frequently asked questions
Is 2¹⁰ the same as "2 multiplied by itself 10 times"?
Not quite, which is why the calculator avoids that phrase. The exponent counts factors, not multiplications. 2¹⁰ has 10 factors of 2 joined by 9 multiplication signs. If you start with 2 and multiply by 2 ten more times, you get 11 factors, which is 2¹¹ = 2,048. What it means therefore reads "10 factors of 2" for 2¹⁰ = 1,024.
Why does the calculator say 0⁰ is undefined?
Because two patterns disagree. Every nonzero number to the power 0 is 1, which suggests 0⁰ = 1; but 0 to any positive power is 0, which suggests 0⁰ = 0. Different books and programs settle this differently, so the calculator shows "Undefined" and explains: 0⁰ is left undefined here (many algebra texts define it as 1 by convention). JavaScript's own Math.pow(0, 0), for example, returns 1. If your course or textbook uses a convention, follow it.
Why is any number to the power 0 equal to 1?
Divide a power by itself: 2⁵ ÷ 2⁵ = 1, and by the quotient rule 2⁵ ÷ 2⁵ = 2⁵⁻⁵ = 2⁰. So 2⁰ must be 1. The argument works for any nonzero base, which is why 1000⁰ and (−5)⁰ both give 1 and What it means reads "Any nonzero number to the power 0 is 1". It fails for 0, because 0 ÷ 0 is undefined.
Why can't I raise 0 to a negative power?
A negative exponent means a reciprocal: 0⁻¹ = 1 ÷ 0¹ = 1 ÷ 0, which is division by zero. The calculator shows "Undefined" and "0 to a negative power would mean dividing by 0". Positive powers of 0 are fine: 0⁵ = 0.
Why do some negative bases with a decimal exponent say "Not a real number"?
A decimal exponent stands for a fraction, and the denominator is a root. The calculator writes the decimal exactly as a fraction in lowest terms. With an odd denominator the root is real: 0.2 = 1/5, so (−32)0.2 = −2, and 0.6 = 3/5, so (−27)0.6 ≈ −7.22467. With an even denominator it is not: (−8)0.5 would be √(−8), which is not a real number. A typed decimal like 0.333333 is 333333/1000000, not exactly 1/3, and that denominator is even, so the calculator does not guess. For odd roots of negative numbers, use the square root calculator with n = 3, 5 and so on.
How do I read the scientific notation?
1.000000 × 10^9 means 1 × 10⁹ = 1,000,000,000, and 9.536743 × 10^-7 means 9.536743 ÷ 10,000,000 = 0.0000009536743. The "^" stands for the exponent. Grade 8 standard 8.EE.A.3 describes the same idea as a single digit times an integer power of 10; the calculator keeps 6 decimal places for more precision.
What is the largest result the calculator can show?
About 1.8 × 10³⁰⁸. Beyond that, for example 2¹⁰²⁴, the calculator shows "Too large to display". 2¹⁰²³ still works and is 8.988466 × 10^307. Results become less exact before that point: only about 16 significant digits are kept, so whole-number results above 2⁵³, about 9 × 10¹⁵, can be rounded, as Example 5 shows. Extremely small results, such as 10⁻⁴⁰⁰, are shown as 0.
How are exponents used outside math class?
Repeated growth by the same factor is an exponent. Money growing 5% a year for 10 years is multiplied by 1.05¹⁰ = 1.62889463, so $1,000 becomes about $1,628.89 if no money is added or taken out. The compound interest calculator handles deposits and different compounding periods. Powers of 2 also count binary values: 2¹⁰ = 1,024.