Exponent Calculator

About the Exponent Calculator

Exponents are shorthand for repeated multiplication, but they quickly go further: negative exponents describe reciprocals, fractional exponents describe roots, and a handful of rules let you simplify long products in a few lines. Mixing those rules up is one of the most common sources of algebra mistakes.

The calculator has two modes. Power mode evaluates $a^b$ exactly for rational base and exponent, for example $8^{2/3} = 4$ or $\left(\tfrac{2}{3}\right)^{-2} = \tfrac94$ , and tells you when the result is irrational or not a real number. Simplify mode applies the laws of exponents to monomial expressions with several variables and writes the answer with positive exponents.

What is an exponent?

In $a^n$ , the base $a$ is multiplied by itself $n$ times when $n$ is a positive integer: $2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32$ . The exponent $n$ is also called the power or index.

The definition is extended so that the rules keep working: $a^0 = 1$ for $a \ne 0$ , $a^{-n} = 1/a^n$ , and $a^{1/n} = \sqrt[n]{a}$ . With these, every rational exponent has a meaning, and $a^{m/n}$ is the $n$ -th root of $a$ raised to the power $m$ .

How to use the Exponent Calculator

Choose Power to evaluate a number raised to an exponent, or Simplify to apply exponent laws to an expression.
In Power mode, enter the base and the exponent. Both may be integers, decimals or fractions such as 2/3 or -1/2.
In Simplify mode, type the expression with ^ for powers, for example (2x^3y)^2/(4x^-1y^5).
Press Calculate and follow the steps; each one names the rule used, such as the power of a product or the quotient rule.

Formulas

Product rule

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

Quotient rule

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

Power of a power

(a^m)^n = a^{mn}

Power of a product

(ab)^n = a^n b^n

Negative exponent

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

Fractional exponent

a^{m/n} = \left(\sqrt[n]{a}\right)^m

Worked example: 8^(2/3)

The worked example evaluates $8^{2/3}$ . The denominator 3 of the exponent means a cube root and the numerator 2 means a square, so the engine takes $\sqrt[3]{8} = 2$ first and then squares it to get 4.

Taking the root first keeps the numbers small: $8^2 = 64$ and $\sqrt[3]{64} = 4$ gives the same answer with more arithmetic.

The solution below is generated by the same engine that powers the calculator above.

Final answer

4

How we read your problem

Raise 8 to the power 2/3.

\left(8\right)^{\frac{2}{3}}

Step-by-step solution

Identify the base and exponent
base b = 8, exponent n = 2/3
$8^{\frac{2}{3}}$
Fractional exponent rule
b^(m/n) = (ⁿ√b)^m: the denominator 3 is the root, the numerator 2 is the power.
$8^{\frac{2}{3}} = \left(\sqrt[3]{8}\right)^{2}$
Take the cube root exactly
$4$

Formulas used

Integer power
$b^n = \underbrace{b\cdot b\cdots b}_{n}$
Negative exponent
$b^{-n} = \frac{1}{b^n}$
Rational exponent
$b^{m/n} = \left(\sqrt[n]{b}\right)^m$

8 raised to 2/3 equals 4.

Verification

Decimal check: 8^0.6666666667 ≈ 4.

Detailed explanation

Why the rules work

The product rule is counting. $a^3 \cdot a^2$ is three copies of $a$ multiplied by two more copies, five in total, so $a^5$ . The quotient rule is cancelling: in $\frac{a^5}{a^2}$ , two copies cancel and three remain.

The zero and negative exponent definitions are forced by the quotient rule. Since $\frac{a^3}{a^3} = 1$ and the rule gives $a^0$ , we must have $a^0 = 1$ . Likewise $\frac{a^2}{a^5} = a^{-3}$ and also equals $\frac{1}{a^3}$ , so $a^{-3} = \frac{1}{a^3}$ .

Fractional exponents and roots

Because $(a^{1/n})^n = a^{1} = a$ , the number $a^{1/n}$ must be the $n$ -th root of $a$ . The calculator returns an exact result when the root is rational, such as $32^{3/5} = 8$ , and otherwise keeps the exact radical form instead of a rounded decimal.

For a negative base, roots with an odd index are real, so $(-8)^{1/3} = -2$ , but even-index roots are not real numbers: $(-4)^{1/2}$ has no real value. The calculator reports this instead of returning a misleading result.

Simplifying monomial expressions

To simplify an expression such as $\frac{(2x^3y)^2}{4x^{-1}y^5}$ , first apply the power of a product: $(2x^3y)^2 = 4x^6y^2$ . Then divide coefficients and subtract exponents variable by variable: $4/4 = 1$ , $x^{6-(-1)} = x^7$ and $y^{2-5} = y^{-3}$ . Finally rewrite negative exponents as reciprocals to get $\frac{x^7}{y^3}$ .

Where exponents appear outside algebra class

Exponents describe anything that grows or shrinks by a fixed factor. Money earning 5% a year is multiplied by $1.05^t$ after $t$ years, a population that doubles every period grows like $2^t$ , and a radioactive sample with a half-life shrinks like $\left(\tfrac12\right)^t$ . Scientific notation, such as $6.02 \times 10^{23}$ , uses powers of ten to write very large and very small numbers compactly.

In each case the same laws apply. Two years of growth followed by three more is $1.05^2 \cdot 1.05^3 = 1.05^5$ , and the time needed to reach a target amount is found by undoing the exponent with a logarithm, which the logarithm calculator does step by step.

Common mistakes

Multiplying the base by the exponent, as in $2^3 = 6$ instead of 8.
Thinking a negative exponent makes the result negative. $2^{-3} = \tfrac18$ , which is positive.
Applying the exponent to only part of a product: $(3x)^2 = 9x^2$ , not $3x^2$ .
Distributing an exponent over a sum: $(a + b)^2 \ne a^2 + b^2$ .
Confusing $-3^2 = -9$ with $(-3)^2 = 9$ ; the exponent binds before the minus sign.

Tips

For $a^{m/n}$ , take the root first when the base is a perfect power; the numbers stay smaller.
Handle coefficients and each variable separately when simplifying monomials.
Leave negative exponents until the last step, then move them across the fraction bar.
Use brackets whenever you raise a negative number or a fraction to a power.

Frequently asked questions

What is any number to the power of zero?

Any non-zero number to the power zero equals 1. The expression 0 to the power 0 is left undefined in this calculator because different areas of mathematics treat it differently.

How do I calculate a negative exponent?

Take the reciprocal of the base and use the positive exponent. For example 5 to the power minus 2 equals 1 over 25.

What does a fractional exponent mean?

The denominator is a root and the numerator is a power. So 27 to the power 2/3 is the cube root of 27, which is 3, squared, which is 9.

Can I raise a negative number to a fractional power?

Only when the reduced denominator of the exponent is odd. Minus 8 to the power 1/3 is minus 2, but minus 4 to the power 1/2 is not a real number.

Why is (a + b) squared not a squared plus b squared?

Exponents distribute over multiplication, not addition. Expanding gives a squared plus 2ab plus b squared, and the middle term is easy to forget.

Does the calculator give exact answers?

Yes. Results are exact fractions or radicals; a decimal approximation is shown only alongside an irrational result.

Work out a power

Enter a base and exponent, or a whole expression, above to see the exponent rules applied step by step.

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