Exponent Calculator
Calculate a base raised to an exponent, or find a missing base or exponent. Choose what to solve for, then enter the two known values.
Calculate a Power or Find a Missing Value
Exponent Formula
Where:
- \(a\) — the base
- \(b\) — the exponent
- \(r\) — the result, \(a^b\)
The first equation calculates a power. The rearranged equations find a base or an exponent, subject to the real-number restrictions explained below; in particular, \(b=\ln(r)/\ln(a)\) requires \(a>0\), \(a\ne1\), and \(r>0\).
Exponent – Calculation Example
Example: using the default values, raise the base 2 to the exponent 10.
\[r=2^{10}=2\times2\times2\times2\times2\times2\times2\times2\times2\times2=1024\]
Result: \(1024\), or \(1.024\times10^3\) in scientific notation. Powers are dimensionless unless the base has units; if it does, those units are raised to the same exponent.
How to use the calculator
- In Solve for, choose Result (base raised to exponent), Base, or Exponent. The field for the unknown value is hidden.
- To calculate a power, enter Base (a) and Exponent (b). The default entries, 2 and 10, give 1024.
- To find a base, enter Exponent (b) and Result (a^b). To find an exponent, enter Base (a) and Result (a^b).
- Read Answer for the requested value. Check Exact value, Scientific notation, and Steps and special cases for additional detail or an explanation of why an input has no unique real-number answer.
The Exponent (b) field accepts an integer such as 10, a decimal with at most nine places such as -2.5, or a fraction such as 2/3. Enter the fraction directly rather than a rounded decimal when its denominator matters: the calculator reduces the fraction before checking whether a negative base has a real result.
What an exponent means
An exponent describes a power of a base. For a positive integer exponent, \(a^b\) multiplies \(a\) by itself \(b\) times. A zero exponent gives \(a^0=1\) for any nonzero base. A negative exponent takes a reciprocal: \(a^{-b}=1/a^b\), provided \(a\ne0\). Thus \(2^{-3}=1/8\), not \(-8\).
A fractional exponent combines roots and powers. For integers \(p\) and nonzero \(q\), \(a^{p/q}=(\sqrt[q]{a})^p\) wherever that root is real. For example, \(8^{2/3}=(\sqrt[3]{8})^2=4\). The denominator determines the root; the numerator determines the power. For more direct root calculations, use the Nth Root calculator.
Finding the exponent reverses the operation. If \(a^b=r\), taking logarithms gives \(b=\ln(r)/\ln(a)\). For example, \(2^b=1024\) gives \(b=10\). This logarithm method works here only for a positive base other than 1 and a positive result. The Logarithm calculator can help with logarithms independently of a power equation.
Finding the base can produce more than one answer. If \(a^2=16\), both \(4\) and \(-4\) work, so the answer is \(\pm4\). The calculator accounts for both signs when the reduced exponent has an even numerator and an odd denominator. Otherwise, do not assume that applying a root always produces a negative as well as a positive real base.
Real-number limits and result displays
- Zero: The calculator treats \(0^0\) as undefined. Zero raised to a positive exponent is zero; zero raised to a negative exponent would require division by zero. A nonzero base raised to zero is 1, so solving \(a^0=1\) does not identify a unique base.
- Negative bases: A reduced fractional exponent with an even denominator has no real result for a negative base. An odd denominator can give a real result: \((-8)^{1/3}=-2\). To solve for an unknown exponent, this calculator uses real logarithms and therefore requires a positive base.
- Base 1: Every real exponent makes \(1^b=1\). If the requested result is 1, the exponent is not unique; for any other result, no exponent works.
- Output precision: Exact value is available for supported integer and rational powers of integer bases, subject to a 2,000-digit display limit. Other roots or logarithmic answers are numerical. Scientific notation makes very large or very small answers easier to read; the Scientific Notation calculator offers a separate way to work with that format.
These results concern ordinary real-number powers, not modular arithmetic. If you need the remainder of an integer power after division by another integer, use the Modular Exponentiation calculator instead.
Frequently Asked Questions
How can I calculate a large power by hand more quickly?
Use repeated squaring instead of multiplying by the base one step at a time. For example, to find \(3^{10}\), calculate \(3^2=9\), \(3^4=81\), and \(3^8=6561\). Then \(3^{10}=3^8\times3^2=59{,}049\). This method is especially useful when the exponent is a large whole number.
Is \(-2^4\) the same as \((-2)^4\)?
No. By the usual order of operations, \(-2^4\) means \(-(2^4)=-16\), while \((-2)^4=16\). Parentheses make the negative sign part of the base. If you want to raise a negative number to a power, enter the negative number as the base rather than negating the result afterward.
Why should I enter \(1/3\) instead of \(0.333333333\) for a negative base?
The decimal is only an approximation to \(1/3\), not the same exponent. As an exact fraction, \(0.333333333=333333333/1000000000\), which has an even denominator after reduction. That makes a negative base raised to this exact decimal exponent non-real, even though \((-8)^{1/3}=-2\). Enter the intended fraction to preserve its root.
When can I add or multiply exponents?
Add exponents when multiplying powers with the same base: \(a^m a^n=a^{m+n}\). Multiply exponents when raising a power to another power: \((a^m)^n=a^{mn}\), where the expressions are defined. Neither rule lets you add exponents across different bases: \(2^3\times3^4\) cannot be simplified to \(6^7\).
How do units change when I raise a measurement to a power?
Raise the units along with the number. For example, squaring \(2\,\mathrm{m}\) gives \(4\,\mathrm{m}^2\). If you convert first, \(2\,\mathrm{m}=200\,\mathrm{cm}\), so the same area is \(200^2=40{,}000\,\mathrm{cm}^2\). Remember that area conversions are squared: \(1\,\mathrm{m}^2=10{,}000\,\mathrm{cm}^2\), not \(100\,\mathrm{cm}^2\).
How can I estimate the number of digits in a large power?
For a positive integer exponent \(n\) and an integer base \(a>1\), the digit count of \(a^n\) is \(\lfloor n\log_{10}(a)\rfloor+1\). For example, \(2^{1000}\) has \(\lfloor1000\log_{10}(2)\rfloor+1=302\) digits. If the logarithm is extremely close to a whole number, use more precision before trusting the estimate.
Why might the displayed decimal differ from an exact power?
Ordinary decimal calculations have limited precision, so a numerical result may be rounded even when the mathematical answer is exact. Check the Exact value field when it is available; scientific notation is a compact display format, not necessarily an exact value. Extremely large or small results may also fall outside the range used for ordinary numerical calculations.