Exponent Calculator (Powers, a to the nth Power)

Fill in only the two you know out of base a, exponent n and result y. The third is calculated for you (to find "a to the nth power", fill in the base and the exponent). The equation below is linked to the fields, so you can also type the base, exponent or result right into it.

Base a (number to multiply)

Exponent n (times to multiply)

Result y (enter to solve backward)

Enter numbers only. The exponent can be negative (-3) or a decimal (0.5). The base can also be "e" for Euler's number. Filling in all three gives an error.

Result

Enter the base and the exponent (or the two values you know) in the fields on the left and press "Calculate". The result will appear here.

[ CLOSE ]

What you can do on this page
What is this calculation used for?
How to Use
Formula
Symbols and terms
- Symbols
- Terms
Good to know before you start
How to calculate it in Excel
How to calculate it in Google Sheets
How to calculate it in Python
How to write it in LaTeX and other math languages (copy and paste)
How to have ChatGPT do the calculation
DataChef Features
Related Features
NumberChef Calculators List

What you can do on this page

Enter the base \(a\) and the exponent \(n\), and the power \(a^n\) (\(a\) to the \(n\)th power) appears on the spot
Negative exponents (\(2^{-3}\)), decimal exponents (\(4^{0.5}\)) and negative bases (\((-2)^5\)) work too. To use Euler's number as the base, just type "e"
It also works backward: "2 to what power is 1024?" (solve for the exponent) or "what number cubed is 125?" (solve for the base)
Large results such as \(2^{60} = 1152921504606846976\) are shown with every digit, not rounded off
A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page

Enter the two values you know out of the base, the exponent and the result, and the third is calculated. A negative number with a decimal exponent (for example, (-4) to the 0.5 power) cannot be calculated, because the answer is not a real number.

What is this calculation used for?

Seeing how fast money grows with compound interest (savings, investing and loans)

At 3% interest a year, after 10 years your money grows to \(1.03^{10} \approx 1.344\) times the original (about 1.34 times). Compound interest, where interest earns more interest, is exactly a power: the same growth factor multiplied again and again.
A loan or credit card balance grows by the same formula. This is the power you meet most often in everyday life, and it helps you plan your money with numbers instead of guesses.

Estimating how fast bacteria multiply (food safety)

Under good conditions, some bacteria divide about once every 20 minutes. In 3 hours that is 9 divisions, so 1 bacterium becomes \(2^{9} = 512\) (a rough guide under ideal food and temperature conditions).
This explosive growth of a power is the reason food should not be left out at room temperature.

Calculating how medicines and radioactive materials decrease (medicine and safety)

The amount of a medicine in the body halves with each half-life. After 5 half-lives, \(0.5^{5} = 0.03125\) (about 3%) is left (the real rate differs from person to person and with health).
Radioactive decay follows exactly the same formula. It is the basis for estimating "how much is left" in medicine and emergency planning.

Computers and data sizes (powers of 2)

Computers work with combinations of two choices, 0 and 1 (bits), so powers of 2 appear everywhere. 1 KB has traditionally been \(2^{10} = 1024\) bytes (by the SI standard it is 1000 bytes), 32 bits can store \(2^{32} = 4294967296\) (about 4.3 billion) patterns, and a full-color image has \(2^{24}\) (about 16.78 million) colors.
Engineers and game developers use these powers every day.

Reading a one-step difference in earthquake magnitude (disaster safety)

An earthquake 1 magnitude higher releases \(10^{1.5} \approx 31.6\) times (about 32 times) the energy, and one 2 magnitudes higher releases \(10^{3} = 1000\) times.
Powers show why a magnitude 8 is far more destructive than a magnitude 7, even though the numbers differ by only 1. This helps you read disaster news correctly.

Formula

Power (\(a\) to the \(n\)th power)

Standard notation (the usual math form)

\(y\) \(=\) \(a\) \(n\)

In words (symbols replaced with words)

③ \(y\): result \(=\) ① \(a\): base ② \(n\): exponent

The formula in words

① Take the \(a\): base

② multiply it by itself as many times as the \(n\): exponent says

③ and you get the \(y\): result

Quick example

2 to the 5th power (2 multiplied 5 times) is

\(y\): result \(=\) base (2) exponent (5 times)

\(2^{5} = 2 \times 2 \times 2 \times 2 \times 2 = 32\)

Key idea

The exponent is the "number of times to multiply", written small at the upper right of the number. Also remember two rules: \(a^1 = a\) (using it once leaves it as is) and \(a^0 = 1\) (any nonzero number to the power 0 is defined as 1). With these, negative and fractional exponents in the next formulas follow naturally.

Negative exponent (\(a\) to the power \(-n\))

Standard notation (the usual math form)

\(a^{-n}\) \(=\) \(1\) \(\div\) \(a^{n}\)

In words (symbols replaced with words)

③ \(a^{-n}\): a to the power −n \(=\) ① \(1\): the starting 1 \(\div\) ② \(a^n\): power

The formula in words

① Take the \(1\): the starting 1

② divide it by the \(a^n\): power (\(a\) multiplied \(n\) times)

③ and you get \(a^{-n}\): a to the power −n (the reciprocal of the power)

Quick example

2 to the power −3 is

power −3 \(2^{-3}\) \(=\) \(1\): the starting 1 \(\div\) 2 cubed (8)

\(2^{-3} = 1 \div 2^{3} = 1 \div 8 = 0.125\)

Key idea

A negative exponent stands for "divide", the opposite of "multiply". It is defined so that \(a^{n} \times a^{-n} = a^{n-n} = a^{0} = 1\), a natural extension that agrees with the law of exponents (Formula 4).

Fractional and decimal exponents (\(a\) to the power \(\frac{1}{n}\))

Standard notation (the usual math form)

\(a^{1/n}\) \(=\) \(\sqrt[n]{a}\)

In words (symbols replaced with words)

② \(a^{1/n}\): a to the power 1/n \(=\) ① \(\sqrt[n]{a}\): nth root of a

The formula in words

① The \(\sqrt[n]{a}\): nth root of a is "the number that gives \(a\) when raised to the \(n\)th power". Written with an exponent, it is

② \(a^{1/n}\): a to the power 1/n (a fractional or decimal exponent is a root)

Quick example

4 to the power 0.5 (since \(0.5 = \frac{1}{2}\), this is the square root) is

4 to the power 0.5 \(4^{0.5}\) \(=\) square root of 4 \(\sqrt{4}\)

\(4^{0.5} = 4^{\frac{1}{2}} = \sqrt{4} = 2\)

Key idea

On a calculator and on this page, enter a fractional exponent as a decimal (0.5 for \(\frac{1}{2}\)). A negative number with a fractional or decimal exponent cannot be calculated, because the answer is not a real number (for example, \((-4)^{0.5}\) would be "a number whose square is \(-4\)", and no real number is).

Law of exponents (multiplying adds the exponents)

Standard notation (the usual math form)

\(a^{n}\) \(\times\) \(a^{m}\) \(=\) \(a^{n+m}\)

In words (symbols replaced with words)

① \(a^n\): power \(\times\) ② \(a^m\): power \(=\) ③ \(a^{n+m}\): power with the exponents added

The formula in words

① Multiply the \(a^n\): power with the same base

② by the \(a^m\): power with the same base and add the exponents to get the

③ \(a^{n+m}\): power

Quick example

2 cubed × 2 to the 4th power is

2 cubed (8) \(\times\) 2 to the 4th (16) \(=\) 2 to the 7th (128)

\(2^{3} \times 2^{4} = 2^{3+4} = 2^{7} = 128\)

Key idea

"2 multiplied 3 times" times "2 multiplied 4 times" is 2 multiplied 7 times in total. That is the idea behind the law of exponents. In the same way, division subtracts the exponents, \(a^{n} \div a^{m} = a^{n-m}\), and a power of a power multiplies them, \((a^{n})^{m} = a^{nm}\).

Solving for the exponent (what power? logarithms)

Standard notation (the usual math form)

\(n\) \(=\) \(\ln y\) \(\div\) \(\ln a\)

In words (symbols replaced with words)

③ \(n\): exponent \(=\) ① \(\ln y\): natural log of the result \(\div\) ② \(\ln a\): natural log of the base

The formula in words

① Take the \(\ln y\): natural log of the result

② divide it by the \(\ln a\): natural log of the base

③ and you get the \(n\): exponent (the power you raise \(a\) to in order to get \(y\))

Quick example

The power of 2 that gives 1024 (the \(n\) in \(2^{n} = 1024\)) is

\(n\): exponent \(=\) natural log of 1024 (about 6.931) \(\div\) natural log of 2 (about 0.693)

\(n = \ln 1024 \div \ln 2 = 10\ \ (2^{10} = 1024)\)

Key idea

Finding "what power" is called taking a logarithm (log). The direct way to write it is \(\log_2 1024\) (the logarithm base 2), but most scientific calculators only have a log key (common logarithm) and an ln key (natural logarithm). So you get the same value as "log of the result ÷ log of the base" (the change of base formula). You can use either log or ln, as long as the top and bottom use the same one.

Solving for the base (the number whose \(n\)th power is \(y\), a root)

Standard notation (the usual math form)

\(a\) \(=\) \(y\) \(1/n\)

In words (symbols replaced with words)

③ \(a\): base \(=\) ① \(y\): result ② \(\frac{1}{n}\): reciprocal of the exponent

The formula in words

① Raise the \(y\): result

② to the power of the \(\frac{1}{n}\): reciprocal of the exponent (that is, take the \(n\)th root)

③ and you get the \(a\): base

Quick example

The number whose cube is 125 (the \(a\) in \(a^{3} = 125\)) is

\(a\): base \(=\) result (125) reciprocal of the exponent (1/3)

\(a = 125^{1/3} = \sqrt[3]{125} = 5\ \ (5 \times 5 \times 5 = 125)\)

Key idea

"The number whose \(n\)th power is \(y\)" (the \(n\)th root of \(y\)) is exactly the fractional or decimal exponent of Formula 3. If the result \(y\) is negative, there is an answer only when the exponent is odd (for example, \((-2)^5 = -32\), so the number whose 5th power is \(-32\) is \(-2\)). An even power can never be negative, so in that case there is no solution.

A power is "the same number (the base) multiplied by itself as many times as the exponent says". A negative exponent gives the reciprocal, and a fractional or decimal exponent gives a root. To work backward, use a logarithm to find the exponent and a root (the power 1/n) to find the base.

Symbols and terms

Symbols

\(a\)	a (the base)	The number that is multiplied by itself (the base). (Example - in 2 to the 5th power, it is 2)
\(n\)	n (the exponent)	How many times to multiply (the exponent). It is written small at the upper right of the number. (Example - in 2 to the 5th power, it is 5)
\(a^n\)	a to the nth power (a to the n)	\(a\) multiplied by itself \(n\) times (a power). (Example - \(2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32\))
\(y\)	y	On this page, the result of the power.
\(\sqrt[n]{a}\)	the nth root of a	The number that gives \(a\) when raised to the \(n\)th power (a root). With an exponent, it is the same as \(a^{1/n}\).
\(\ln y\)	natural log of y	The logarithm with base \(e\) (Euler's number). It tells you "\(e\) to what power gives \(y\)", and is used to solve for an exponent.
\(e\)	e (Euler's number)	A special constant, about 2.71828. It describes continuous growth (the limit of compound interest) and appears all over calculus and statistics. On this calculator you can enter "e" as the base.

Terms

power (exponentiation)	Multiplying the same number by itself several times. The operation is called exponentiation, and its result is called a power. "2 to the 3rd power" and "2 raised to the power of 3" say the same thing.
base	The number that is multiplied by itself in a power. It is the \(a\) in \(a^n\).
exponent	The small number at the upper right that tells how many times to multiply. It is the \(n\) in \(a^n\). Its meaning is extended to zero, negative and fractional exponents.
law of exponents (laws of exponents, exponent rules)	The set of rules for working with powers, such as \(a^n \times a^m = a^{n+m}\) (multiplying adds the exponents). The rules that the power 0 gives 1 and a negative exponent gives the reciprocal are chosen so that they agree with these laws.
root (nth root)	A number that gives the original number when raised to the \(n\)th power. Square roots and cube roots are examples. It is the same as a fractional or decimal exponent (the power \(1/n\)).
reciprocal	1 divided by the number (the number you multiply by to get 1). The reciprocal of 2 is 1/2. A negative exponent gives the reciprocal of the power.
logarithm (log)	The number that tells "what power of the base gives the target number". It is the tool for working backward to the exponent of a power.
Euler's number	The special constant \(e = 2.71828\ldots\). It is the base of the natural logarithm and shows up wherever there is continuous growth, such as the limit of interest compounded continuously.

Good to know before you start

Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.

Multiplication (Grade 3)	Being able to multiply the same number again and again
Decimals and fractions (Grades 5–6)	Being able to multiply decimals such as \(0.5 \times 0.5\) Being able to switch between a fraction and a decimal, such as \(\frac{1}{2}\) and 0.5
What exponents are (Grade 6)	Knowing that the small number at the upper right (the exponent) is "how many times to multiply", as in \(2^3 = 2 \times 2 \times 2\) Knowing that a negative number to an even power is positive and to an odd power is negative (\((-2)^4 = 16\), \((-2)^5 = -32\))
Square roots (Grade 8)	Being able to find "the number whose square is the original number", as in \(\sqrt{4} = 2\)
Extending exponents and the laws of exponents (Grade 8 and Algebra 2)	Knowing that the power 0 gives 1, a negative exponent gives the reciprocal, and a fractional exponent gives a root Being able to simplify products of powers with the law of exponents \(a^n \times a^m = a^{n+m}\)
Logarithms (Algebra 2)	Knowing that a logarithm (log) finds "what power of the base gives the target number" (used to solve for the exponent)

How to calculate it in Excel

Copy the whole table below and paste it into cell A1 in Excel. It works as is.

Table to find a power (a to the nth power)

Base a	2
Exponent n	5
Power a^n	=B1^B2

Table for a negative exponent

Base a	2
Exponent n (to use as -n)	3
Power -n: a^(-n)	=B1^(-B2)
Same value as the reciprocal 1/a^n	=1/B1^B2

Table for a fractional or decimal exponent (root)

Base a	4
Exponent (as a decimal)	0.5
a^0.5 (same as the square root)	=B1^B2
Check with SQRT	=SQRT(B1)

Table to check the law of exponents (multiplying adds exponents)

Base a	2
Exponent n	3
Exponent m	4
a^n × a^m	=B1^B2*B1^B3
a^(n+m) (same value)	=B1^(B2+B3)

Table to solve for the exponent (what power?)

Base a	2
Result y	1024
Exponent n (a to what power is y)	=LOG(B2,B1)

Table to solve for the base (the number whose nth power is y)

Exponent n	3
Result y	125
Base a (y to the power 1/n)	=B2^(1/B1)

After pasting, column A holds the labels and column B the numbers. The upper rows are your inputs, and the formulas in the last rows calculate from them automatically.
"^" is the power symbol, so "=B1^B2" multiplies the value in B1 by itself B2 times. To solve for an exponent, use the LOG function (=LOG(number, base)).
For example, B3 shows 32 in the first table and 10 in the fifth table. Just replace the input numbers with your own.

How to calculate it in Google Sheets

Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.

Table to find a power (a to the nth power)

Base a	2
Exponent n	5
Power a^n	=B1^B2

Table for a negative exponent

Base a	2
Exponent n (to use as -n)	3
Power -n: a^(-n)	=B1^(-B2)
Same value as the reciprocal 1/a^n	=1/B1^B2

Table for a fractional or decimal exponent (root)

Base a	4
Exponent (as a decimal)	0.5
a^0.5 (same as the square root)	=B1^B2
Check with SQRT	=SQRT(B1)

Table to check the law of exponents (multiplying adds exponents)

Base a	2
Exponent n	3
Exponent m	4
a^n × a^m	=B1^B2*B1^B3
a^(n+m) (same value)	=B1^(B2+B3)

Table to solve for the exponent (what power?)

Base a	2
Result y	1024
Exponent n (a to what power is y)	=LOG(B2,B1)

Table to solve for the base (the number whose nth power is y)

Exponent n	3
Result y	125
Base a (y to the power 1/n)	=B2^(1/B1)

The same formulas as in Excel (the ^ operator and the LOG and SQRT functions) work as is. Copy the whole table, paste it into cell A1, and replace the input numbers with your own.

How to calculate it in Python

base = 2       # base (the number multiplied)
exponent = 5   # exponent (how many times to multiply)

power = base ** exponent   # power (base to the exponent)
print(f"{base} to the power of {exponent}: {power}")

# --- To work backward (logarithms and roots), use the math module ---
import math

n = math.log(1024, 2)   # solve for the exponent: 2 to what power is 1024
a = 125 ** (1 / 3)      # solve for the base: the number whose cube is 125 (cube root)
print(f"2 to what power is 1024: {n}")
print(f"The number whose cube is 125: {round(a, 10)}")

Runs with the standard library only. "**" is the power operator. Negative exponents (2 ** -3) and decimal exponents (4 ** 0.5) can be written as is. Solving for the base can be off by a tiny rounding error (giving 4.999…), so the result is rounded with round.

How to write it in LaTeX and other math languages (copy and paste)

Power (\(a\) to the \(n\)th power)

y = aⁿ

y = a^{n}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>y</mi>
    <mo>=</mo>
    <msup><mi>a</mi><mi>n</mi></msup>
  </mrow>
</math>

y = a^n

a^n

y := a^n;

y = a^n;

y = a^n

Negative exponent (\(a\) to the power \(-n\))

a⁻ⁿ = 1/aⁿ

a^{-n} = \frac{1}{a^{n}}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <msup><mi>a</mi><mrow><mo>&#x2212;</mo><mi>n</mi></mrow></msup>
    <mo>=</mo>
    <mfrac><mn>1</mn><msup><mi>a</mi><mi>n</mi></msup></mfrac>
  </mrow>
</math>

a^(-n) = 1/a^n

a^(-n)

y := 1/a^n;

y = 1/a^n;

a^(-n) = 1/a^n

Fractional and decimal exponents (\(a\) to the power \(\frac{1}{n}\))

a^(1/n) = ⁿ√a

a^{1/n} = \sqrt[n]{a}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <msup><mi>a</mi><mrow><mn>1</mn><mo>/</mo><mi>n</mi></mrow></msup>
    <mo>=</mo>
    <mroot><mi>a</mi><mi>n</mi></mroot>
  </mrow>
</math>

a^(1/n) = root(n)(a)

Surd[a, n]

y := surd(a, n);

y = nthroot(a, n);

a^(1/n) = √(n&a)

Law of exponents (multiplying adds the exponents)

aⁿ × aᵐ = aⁿ⁺ᵐ

a^{n} \times a^{m} = a^{n+m}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <msup><mi>a</mi><mi>n</mi></msup>
    <mo>&#xD7;</mo>
    <msup><mi>a</mi><mi>m</mi></msup>
    <mo>=</mo>
    <msup><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>m</mi></mrow></msup>
  </mrow>
</math>

a^n xx a^m = a^(n+m)

a^n*a^m

y := a^n*a^m;

y = a^n*a^m;

a^n × a^m = a^(n+m)

Solving for the exponent (what power? logarithms)

n = ln y ÷ ln a

n = \frac{\ln y}{\ln a}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>n</mi>
    <mo>=</mo>
    <mfrac>
      <mrow><mi>ln</mi><mo>&#x2061;</mo><mi>y</mi></mrow>
      <mrow><mi>ln</mi><mo>&#x2061;</mo><mi>a</mi></mrow>
    </mfrac>
  </mrow>
</math>

n = ln(y)/ln(a)

Log[a, y]

n := log[a](y);

n = log(y)/log(a);

n = ln(y)/ln(a)

Solving for the base (the number whose \(n\)th power is \(y\), a root)

a = y^(1/n) = ⁿ√y

a = y^{1/n} = \sqrt[n]{y}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>a</mi>
    <mo>=</mo>
    <msup><mi>y</mi><mrow><mn>1</mn><mo>/</mo><mi>n</mi></mrow></msup>
    <mo>=</mo>
    <mroot><mi>y</mi><mi>n</mi></mroot>
  </mrow>
</math>

a = y^(1/n) = root(n)(y)

Surd[y, n]

a := surd(y, n);

a = nthroot(y, n);

a = y^(1/n) = √(n&y)

How to have ChatGPT do the calculation

You are a calculation assistant for powers and exponents. Do the following calculations by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing).

1. Find 2 to the 5th power.
2. Find the power of 2 that gives 1024 (the exponent).
3. Find the number whose cube is 125 (the base).

Show the formulas you used and the numbers from the execution result for each.

How to Use

1

Enter your numbers

Type the numbers you want to calculate with into the input fields
2

Calculate

Press the "Calculate" button
3

Check the result

The result appears on the spot. The same page also explains the idea behind the calculation and the formula